Goran swap 2 kilometers against the current in the same amount of time it took him to swim 8 kilometers with the current. The rate of the current was 3 kilometers per hour. How fast would Goran swim if there were no current?

Answers

Answer 1

Goran's swimming speed in still water is 5 km/h.

We have,

Let's call Goran's swimming speed in still water s.

When Goran is swimming against the current, his effective speed (relative to the ground) is:

= s - 3 km/h

(since the current is flowing against him),

And when he is swimming with the current, his effective speed is:

= s + 3 km/h.

The time it takes Goran to swim 2 kilometers against the current is the same as the time it takes him to swim 8 kilometers with the current.

Using the formula,

Time = distance/speed,

2 / (s - 3) = 8 / (s + 3)

Solve for s

2(s + 3) = 8(s - 3)

2s + 6 = 8s - 24

6s = 30

s = 5

Therefore,

Goran's swimming speed in still water is 5 km/h.

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Related Questions

A car manufacturer provider information about two different car models. The graph and table each show a proportional relationship between the number of miles traveled and the advertised number of gallons of gas used for two car models.

Answers

Car A can travel 1.25 times the distance car B can travel when both cars use 1 gallons of gas.

What is a proportional relationship?

In Mathematics and Geometry, a proportional relationship refers to a type of relationship that produces equivalent ratios and it can be modeled or represented by the following mathematical equation:

y = kx

Where:

y represents the distance.x represents the gas (gallons).k is the constant of proportionality.

Next, we would determine the constant of proportionality (k) for car A by using various data points as follows:

Constant of proportionality, k = y/x

Constant of proportionality, k = 50/2

Constant of proportionality, k = 25.

y = 25x = 25(1) = 25 miles.

For car B, the constant of proportionality (k) is given by;

Constant of proportionality, k = 60/3

Constant of proportionality, k = 20.

y = 20x = 20(1) = 20 miles.

Difference = 25/20 = 1.25.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

mateo needs to rent a car for one day. he can rent a subaru from starry car rental for $31.19 per day plus 47 cents per mile. he can get the same car from ocean car rental for $48.57 per day plus 36 cents per mile. how much will he pay when starry and ocean will cost the same? this is a money amount so round the answer to the nearest cent.

Answers

Mateo will pay $56.32 when Starry and Ocean car rentals will cost the same.

Let's start by defining the cost functions for both car rentals. For Starry car rental, the cost function can be expressed as C1 = 31.19 + 0.47m, where m is the number of miles driven. For Ocean car rental, the cost function can be expressed as C2 = 48.57 + 0.36m.

We want to find the point where C1 = C2, so we can set the two cost functions equal to each other and solve for m:

31.19 + 0.47m = 48.57 + 0.36m

0.11m = 17.38

m = 158

So when Mateo drives 158 miles, the cost of renting from Starry car rental and Ocean car rental will be the same. We can then substitute m = 158 into either cost function to find the cost:

C1 = 31.19 + 0.47(158) = $107.33

C2 = 48.57 + 0.36(158) = $107.33

Therefore, Mateo will pay $107.33 to rent from either car rental when he drives 158 miles. However, we need to find the cost for just one day of rental. To do this, we can subtract the fixed daily cost from each cost function:

C1 = 31.19(1) + 0.47(158) = $105.33

C2 = 48.57(1) + 0.36(158) = $105.33

So, when Mateo rents a car for one day and drives 158 miles, he will pay $105.33 from either car rental. However, this is not the final answer as we need to find the cost when both car rentals will cost the same. To do this, we can substitute m = 158 into either cost function and round the result to the nearest cent:

C1 = 31.19(1) + 0.47(158) = $105.33

C2 = 48.57(1) + 0.36(158) = $105.33

Therefore, Mateo will pay $56.32 when Starry and Ocean car rentals will cost the same.

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Marlon has $600 in his bank account and plans to withdraw $25 each week. Cassandra has $45 in her bank account and plans to deposit $12 each week. Which equation can be used to determine the number of weeks it will take until Marlon and Cassandra have the same balance in their accounts? A)
600−25x=45+12x


cross out

B)
600x−25=45x+12


cross out

C)
25x+12x=600+45


cross out

D)
600x−45x=25+12


Part B

Answers

As per the given situation, it will take 15 weeks for Marlon and Cassandra to have the same balance in their accounts is 600−25x=45+12x. The correct option is A.

We can start by setting up an equation that represents the balance of each person after x weeks.

After x weeks, Marlon's balance would be:

600 - 25x

After x weeks, Cassandra's balance would be:

45 + 12x

To find the number of weeks it will take until they have the same balance, we can set these two expressions equal to each other:

600 - 25x = 45 + 12x

Simplifying this equation, we can combine like terms:

600 = 45 + 37x

Subtracting 45 from both sides:

555 = 37x

Dividing both sides by 37:

x = 15

Therefore, it will take 15 weeks for Marlon and Cassandra to have the same balance in their accounts.

Thus, the correct equation is A) 600−25x=45+12x.

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Let f: R→R be a continuous function such that f(R) ⊂ Q. Show that f is constant.

Answers

[tex]$f(x) = q_a$[/tex] for all [tex]$x \in (a-1, a+1)$[/tex]. Since a was arbitrary, it follows that f is constant on any open interval. Since f is continuous, it follows that f is constant on [tex]$\mathbb{R}$[/tex].

Let a be any real number in R. Since f is continuous, the intermediate value theorem implies that the image of any closed interval under f is also an interval. Therefore, [tex]$f([a-1, a+1])$[/tex] is an interval in [tex]$\mathbb{Q}$[/tex]. Since the only intervals in [tex]$\mathbb{Q}$[/tex] are single points, [tex]$f([a-1, a+1]) = {q_a}$[/tex] for some rational number [tex]$q_a$[/tex].

Now let b be any real number with [tex]$b > a+1$[/tex]. By the intermediate value theorem, there exists some [tex]$x \in [a, b]$[/tex] such that [tex]$f(x) = \frac{q_a+q_b}{2}$[/tex]. But since f takes only rational values, [tex]$f(x) = q_a$[/tex]. This argument applies to all real numbers b with [tex]$b > a+1$[/tex], so [tex]$f(x) = q_a$[/tex] for all [tex]$x > a+1$[/tex]. Similarly, we can show that [tex]$f(x) = q_a$[/tex] for all [tex]$x < a-1$[/tex].

Therefore, [tex]$f(x) = q_a$[/tex] for all [tex]$x \in (a-1, a+1)$[/tex]. Since a was arbitrary, it follows that f is constant on any open interval. Since f is continuous, it follows that f is constant on [tex]$\mathbb{R}$[/tex].

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Write the general form equation for the circle shown.

Answers

Check the picture below.

so the circle has a radius of 3 and a center at (-2 , 1)

[tex]\textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \hspace{5em}\stackrel{center}{(\underset{-2}{h}~~,~~\underset{1}{k})}\qquad \stackrel{radius}{\underset{3}{r}} \\\\[-0.35em] ~\dotfill\\\\ ( ~~ x - (-2) ~~ )^2 ~~ + ~~ ( ~~ y-1 ~~ )^2~~ = ~~3^2\implies (x+2)^2 + (y-1)^2 = 9[/tex]

Algibra 1, unit 1, Math Nation

Answers

The area of the rectangle is √32 x √45.

Option A is the correct answer.

We have,

From the figure,

Length = √32

Width = √45

Now,

The area of the rectangle.

= Length x width

= √32 x √45

Thus,

The area of the rectangle is √32 x √45.

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One Lump Sum:

Calculate the APR for a $2000 loan that is paid off in one lump sum at the end of the year. The stated annual interest rate is 8%. Show your work.

Answers

The APR for this loan is 8%.

We have,

The formula for calculating APR is:

APR = (r/n) x m

where r is the stated annual interest rate, n is the number of times the interest is compounded in a year, and m is the number of payments made in a year.

In this case,

The loan is paid off in one lump sum at the end of the year, so there is only one payment made in a year (m = 1).

The stated annual interest rate is 8%, so r = 0.08.

We need to determine the value of n.

Since the loan is paid off in one lump sum at the end of the year, we can assume that the interest is compounded annually (n = 1).

Using the formula, we get:

APR = (0.08/1) x 1

APR = 0.08

Therefore,

The APR for this loan is 8%.

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A spinner with repeated colors numbered from 1 to 8 is shown. Sections 1 and 8 are purple. Sections 2 and 3 are yellow. Sections 4, 5, and 6 are blue. Section 7 is red.

Spinner divided evenly into eight sections with three colored blue, one red, two purple, and two yellow.

Determine the theoretical probability of the spinner not landing on blue, P(not blue).

0.375
0.625
0.750
0.875

Answers

The theoretical probability of the spinner not landing on blue would be = 0.625. That is option B.

How to calculate the theoretical probability of the given event?

To calculate the theoretical probability of the given event, the formula that should be used is given as follows:

Probability = possible outcome/sample space

The possible outcome for other colours apart from blue = 5

The sample space = 8

Therefore probability = 5/8 = 0.625

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Which ratio is always equivalent to the sine of angle A?

Answers

Answer: 4 Which ratio represents the cosine of angle A in the right triangle below? ... Which ratio is always equivalent to the sine of ∠A?

Step-by-step explanation:

the domain of function f is (-oo, oo). the value of the function what function could be f

Answers

The function that has the domain (-∞, ∞) which satisfies this condition is f(x) = (x² - 36)/(x - 6)

What is the domain of a function?

The domain of a function is the range of input values to the function.

Given that the domain of a function f is (-∞, ∞). the value of the function what function could be f

To determine the value of the function that satisfies this condition, we look at each of the functions given in  the list.

Now, for each function, that has a linear denominator, the domain has and upper limit which is an integer.The only function which has a domain that does not have an upper limit is the function with a polynomial numerator. This function has a domain of (-∞, ∞).

So, the function which satisfies this condition is f(x) = (x² - 36)/(x - 6)

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can someone help me with this

Answers

the answer is C. Because all your doing is counting how many numbers you see in the parentheses and seeing if it matches the numbers on the graph

!!WILL GUVE BRAINLIEST!!!

Find m UW

Answers

Answer:

The measure of an angle formed by two chords intersecting inside a circle is equal to half the sum of the measures of the intercepted arcs associated with the angle and its vertical angle counterpart.

Let x be the measure of arc UW.

36° = (1/2)(26° + x°)

72° = 26° + x

x = 46°

So the measure of arc UW is 46°.

For the system of equations shown, what is the value of x+y? 3x+y=−6−3x−4y=−12

Answers

Answer:

x + y = -12 + 30 = 18

Step-by-step explanation:

To solve this system of equations, we can use the elimination method. Multiplying the first equation by 3, we get:

9x + 3y = -18

Adding this to the second equation, we eliminate the x terms:

9x + 3y = -18

-3x - 4y = -12

-----------------

-y = -30

Solving for y, we get y = 30. Substituting this back into either equation, we can solve for x:

3x + 30 = -6

3x = -36

x = -12

Therefore, x + y = -12 + 30 = 18.

Let X1, ... , Xn be iid ~ Poisson(1), where l E (0,00) is an unknown parameter. Find the MLE for based on the observations x1 = 2, X2 = 5, X3 = 2, X4 = 1, X5 = 1. =

Answers

Based on the observations, the maximum likelihood estimate of λ is 11/5.

What is probability?

Probability is a field of mathematics that calculates the likelihood of an experiment occurring. We can know everything from the chance of getting heads or tails in a coin to the possibility of inaccuracy in study by using probability.

The probability mass function (PMF) of a Poisson distribution with parameter λ is given by:

P(X = k) = [tex](e^{(-\lambda)} * \lambda^k)[/tex] / k!

The likelihood function for a sample of size n from a  is given by:

L(λ) = P(X1 = x1, X2 = x2, ..., Xn = xn) = ∏[i=1 to n] [tex]( e^{(-\lambda)} * \lambda^{xi})[/tex] / xi!

The log-likelihood function is then:

ln L(λ) = ln ∏[i=1 to n] [tex](e^{(-\lambda)} * \lambda^{xi})[/tex] / xi! = ∑[i=1 to n] [tex](ln e^{(-\lambda)} * \lambda^{xi})[/tex] - ∑[i=1 to n] ln(xi!)

Simplifying further, we get:

ln L(λ) = (-nλ) + (∑[i=1 to n] xi)ln(λ) - ∑[i=1 to n] ln(xi!)

To find the maximum likelihood estimate (MLE) of λ, we need to differentiate the log-likelihood function with respect to λ, set the derivative to zero, and solve for λ.

d/dλ ln L(λ) = -n + (∑[i=1 to n] xi)/λ = 0

Solving for λ, we get:

λ = (∑[i=1 to n] xi) / n

Substituting the given values, we get:

λ = (2 + 5 + 2 + 1 + 1) / 5 = 11 / 5

Therefore, the maximum likelihood estimate of λ, based on the given observations, is 11/5.

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For the following exercises, determine a. Intervals where f is increasing or decreasing, b. Local minima and maxima of f, c. Intervals where f is concave up and concave down, and d. The inflection points of f.

f(x) = x² - 6x

f(x) = x³ - 6x²

f(x) = x⁴ - 6x³

Answers

a.   (-∞,3) - f(x) is decreasing

   (3,∞) - f(x) is rising.

b. Local minima at x=3. No local Minima

c. The function f(x)=x²-6x is always concave upwards.

d. Concave up and does not change concavity, so, No Inflection points.

f(x)= x²-6x

f'(x) = 2x-6

f"(x) = 2

Critical point f'(x)=0

2x-6=0

x=3

Thus, we have two sub intervals over the entire number line. (-∞,3) , (3,∞)

a) sub-interval      x-value           f'(x)                           verdict

         (-∞,3)                1             2(1)-6=-4<0             f(x) is decreasing  

          (3,∞)                4            2(4)-6=2>0              f(x) is increasing

b) At x=3; Before x=3, f(x) decreasing and after x=3, f(x)

is increasing, thus Local minima at x=3.

No local Minima

c) Since f"(x)=2 Always, the function f(x)=x²-6x is always concave upwards

d) Inflection points

Since graph of function f(x)=x²-6x have only been concave up and does not change concavity,

No Inflection points.

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prove that in a group of 250 students, the family name of at least ten students must start with the same letter. there are 26 letters in the english alphabet

Answers

To prove that in a group of 250 students, the family name of at least ten students must start with the same letter, we can use the Pigeonhole Principle.

The Pigeonhole Principle states that if you have n pigeonholes (in this case, the 26 letters of the English alphabet) and you are placing m > n items (here, 250 students) into the pigeonholes, at least one pigeonhole must contain more than one item.

Here's a step-by-step explanation:

1. We have 26 pigeonholes, representing the 26 letters of the English alphabet.

2. We have 250 students (items) to place into these pigeonholes based on the first letter of their family name.

3. Apply the Pigeonhole Principle: Divide the total number of students (250) by the total number of pigeonholes (26).

  250 ÷ 26 ≈ 9.6

4. Since we can't have a fraction of a student, we round up to the nearest whole number.

  10 students per pigeonhole

5. By rounding up, we find that at least one pigeonhole (letter of the alphabet) must have 10 or more students with family names starting with that letter.

So, in a group of 250 students, the family name of at least ten students must start with the same letter, as demonstrated by the Pigeonhole Principle.

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Find the slope of the line through points (4,6) and (-6,2).
044
OB. 215
OD. 52
Beses Selection

Answers

The slope of the line is 2/5 (B)

Gabrielle is writing a thank-you note to a friend. She has 2 kinds of cards and 8 kinds of envelopes that fit the cards. She has 3 designs of first-class stamps, although she only needs to use one. Finally, Gabrielle has to pick a color of pen with which to write the note, and she has 8 to choose from. How many different ways can the thank-you note look?

Answers

There are 384 different ways or combinations that the thank-you note can look.

To find the total number of different ways that the thank-you note can look, we need to multiply the number of choices available for each decision point.

Gabrielle can choose between 2 kinds of cards, so there are 2 options. She has 8 kinds of envelopes to choose from, so there are 8 options. She only needs to use one of the 3 designs of first-class stamps, so there are 3 options. Finally, she has 8 colors of pen to choose from, so there are 8 options.

Therefore, the total number of different ways that the thank-you note can look is:

2 x 8 x 3 x 8 = 384

There are 384 different ways that Gabrielle can create the thank-you note by choosing a card, an envelope, a stamp design, and a pen color.

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Let V be a subspace of Rn with a basis S = {v1..., Vs} and V+ be the subspace of R" orthogo- nal to V with a basis T = {W1,...,Wi}. (a) Prove that S T = Ø.
(b) Prove that SU T forms a basis of Rn
(c) ahow that dim (v)+dim(v1)=n

Answers

S U T is a basis of Rn, and we have

dim(S U T) = dim(S) + dim(T) = s + (n - s) = n

which confirms that S and T are complementary subspaces of Rn.

(a) To prove that S ∩ T = Ø, we need to show that there is no vector that belongs to both S and T.

Assume for contradiction that there exists a vector v that belongs to both S and T. Then, since v is in T, it is orthogonal to all vectors in S, including itself. But since v is in S, it can be expressed as a linear combination of the basis vectors of S, which means it is also not orthogonal to some vector in S, a contradiction. Therefore, S ∩ T = Ø.

(b) To prove that S U T forms a basis of Rn, we need to show that it spans Rn and is linearly independent.

(i) Spanning property: Let x be any vector in Rn. Since S is a basis of V, x can be expressed as a linear combination of the vectors in S. Let y = x - s be the difference between x and the projection of x onto V along S, where s is the projection of x onto V along S. Then y is orthogonal to V, and thus y is in T. Therefore, x = s + y, where s is in V and y is in T. Since s is a linear combination of vectors in S and y is a linear combination of vectors in T, we conclude that S U T spans Rn.

(ii) Linear independence: Assume that there exist scalars c1, c2, ..., cn and d1, d2, ..., dm such that

c1v1 + c2v2 + ... + cnvn + d1w1 + d2w2 + ... + dmwm = 0

where 0 is the zero vector in Rn. We want to show that all the ci's and di's are zero.

Since the vectors in S are linearly independent, we know that c1 = c2 = ... = cn = 0. Thus, the equation reduces to

d1w1 + d2w2 + ... + dmwm = 0

Since the vectors in T are also linearly independent, we know that d1 = d2 = ... = dm = 0. Therefore, S U T is linearly independent.

Since S U T spans Rn and is linearly independent, it forms a basis of Rn.

(c) We know that S is a basis of V, so dim(V) = |S| = s. Let S' be the orthogonal complement of S in Rn, i.e., S' = {x in Rn: x is orthogonal to all vectors in S}. Then, dim(S') = n - s.

We also know that T is a basis of V', the orthogonal complement of V in Rn. Since V and V' are orthogonal complements of each other, we have dim(V) + dim(V') = n. Therefore, we have

dim(T) = dim(V') = n - dim(V) = n - s

Adding the dimensions of S and T, we get

dim(S) + dim(T) = s + (n - s) = n

Therefore, S U T is a basis of Rn, and we have

dim(S U T) = dim(S) + dim(T) = s + (n - s) = n

which confirms that S and T are complementary subspaces of Rn.

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Find the value of x in the given

right triangle.

10

7

х

x = [?]°

Enter your answer as a decimal rounded to the

nearest tenth.

Enter

Answers

In the given case,   x = cos⁻¹  (0.7) ≈ 44.4   ° rounded to the nearest tenth.

To find the value of x in degrees, we can use trigonometric ratios. In a right triangle, the sine of an acute angle is defined as the length of the side opposite the angle divided by the length of the hypotenuse. The cosine of an acute angle is defined as the length of the adjacent side divided by the length of the hypotenuse.

In this case, we have the side opposite to angle x is 7 and the hypotenuse is 10.

Therefore, sin(x) = 7/10. Solving for x, we get x = sin⁻¹ (7/10) ≈ 44.4° rounded to the nearest tenth.

Alternatively, we could use the cosine ratio since we also know the adjacent side. We have the adjacent side as x and the hypotenuse as 10. Therefore, cos(x) = x/10.

Solving for x, we get x = 10cos(x). We also have the opposite side as 7, which means that sin(x) = 7/10. Using the identity sin²(x) + cos²(x) = 1, we can solve for cos(x) as cos(x) = [tex]√(1 - sin²(x)[/tex]). Substituting sin(x) = 7/10, we get cos(x) = √(1 - (7/10)²) ≈ 0.7. Thus, x = cos⁻¹(0.7) ≈ 44.4° rounded to the nearest tenth.

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Find the value of x in the given right triangle.

10⁷х x = [?]°

Enter your answer as a decimal rounded to the  nearest tenth.

Paula is a peach grower in central Georgia and wants to expand her peach orchard, there are 30 trees per acre and the average yield per tree is 600 peaches. Data from the local agricultural experiment station indicates that if Paula wants to plant more than 30 trees per acre, while the trees are in production, the average yield of 600 peaches per tree will be decreased by 12 peaches for each tree over 30. She needs to decide how many trees to plant in the new section of the orchard


A. Is this relationship linear or nonlinear explain your reasoning


B. If Paula plant six more trees per acre, what will be the average yield in peaches per tree? What is the average yield in peaches per tree if she plants 42 trees per acre


Answers

The relationship between the number of trees per acre and the average yield per tree is nonlinear, and Paula needs to balance the trade-off between the number of trees and the average yield per tree when deciding how many trees to plant in the new section of the orchard.

A. The relationship between the number of trees per acre and the average yield per tree is nonlinear. This is because the reduction in yield is not constant and varies based on the number of trees per acre. As per the given information, if Paula plants more than 30 trees per acre, the average yield of 600 peaches per tree will decrease by 12 peaches for each additional tree over 30.

This indicates that the decrease in yield is not linearly proportional to the increase in the number of trees per acre. Instead, the decrease in yield per tree increases as the number of trees per acre increases. Thus, the relationship between the number of trees per acre and the average yield per tree is nonlinear.

B. If Paula plants six more trees per acre, the average yield in peaches per tree will decrease by 12 peaches for each additional tree over 30. Therefore, the average yield per tree would be 588 peaches. On the other hand, if Paula plants 42 trees per acre, there will be 12 additional trees over 30 per acre.

Therefore, the average yield per tree would decrease by 12 peaches for each of these additional trees, which would result in an average yield of 564 peaches per tree. Thus, Paula needs to consider the trade-off between the number of trees and the average yield per tree when deciding how many trees to plant in the new section of the orchard. She should plant the number of trees that maximize her overall yield, considering both the number of trees per acre and the average yield per tree.

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Determine whether the following are subspaces of P4. If so, prove it. If not, show orexplain why. (a. ) The set of all polynomials in P4 of even degree. (b. ) The set of all polynomials of degree 3. (c. ) The set of all polynomials p 2 P4 such that p(0) = 0. (d. ) The set of all polynomials in P4 having at least one real root

Answers

The zero vector in P4 is the polynomial 0(x) = 0, which has even degree. The set of all polynomials in P4 of even degree is closed under addition.

The set of all polynomials in P4 of even degree satisfies all three conditions, it is a subspace of P4.

(a) The set of all polynomials in P4 of even degree is a subspace of P4.

To prove this, we need to show that it satisfies the three conditions for a subspace:

i) It contains the zero vector: The zero vector in P4 is the polynomial 0(x) = 0, which has even degree, so it is contained in the set of all polynomials in P4 of even degree.

ii) It is closed under addition: Let p(x) and q(x) be two polynomials in P4 of even degree. Then, p(x) + q(x) is also a polynomial of even degree, since the sum of two even numbers is even. Therefore, the set of all polynomials in P4 of even degree is closed under addition.

iii) It is closed under scalar multiplication: Let p(x) be a polynomial in P4 of even degree, and let c be a scalar. Then, cp(x) is also a polynomial of even degree, since multiplying an even number by a scalar yields an even number. Therefore, the set of all polynomials in P4 of even degree is closed under scalar multiplication.

Since the set of all polynomials in P4 of even degree satisfies all three conditions, it is a subspace of P4.

(b) The set of all polynomials of degree 3 is not a subspace of P4.

To prove this, we only need to show that it does not satisfy the first condition for a subspace:

i) It contains the zero vector: The zero vector in P4 is the polynomial 0(x) = 0, which has degree 0, not degree 3. Therefore, the set of all polynomials of degree 3 does not contain the zero vector and is not a subspace of P4.

(c) The set of all polynomials p in P4 such that p(0) = 0 is a subspace of P4.

To prove this, we need to show that it satisfies the three conditions for a subspace:

i) It contains the zero vector: The zero vector in P4 is the polynomial 0(x) = 0, which satisfies 0(0) = 0, so it is contained in the set of all polynomials p in P4 such that p(0) = 0.

ii) It is closed under addition: Let p(x) and q(x) be two polynomials in P4 such that p(0) = 0 and q(0) = 0. Then, (p+q)(0) = p(0) + q(0) = 0, so p+q is also a polynomial in P4 such that (p+q)(0) = 0. Therefore, the set of all polynomials p in P4 such that p(0) = 0 is closed under addition.

iii) It is closed under scalar multiplication: Let p(x) be a polynomial in P4 such that p(0) = 0, and let c be a scalar. Then, (cp)(0) = c(p(0)) = c(0) = 0, so cp is also a polynomial in P4 such that (cp)(0) = 0. Therefore, the set of all polynomials p in P4 such that p(0) = 0 is closed under scalar multiplication.

Since the set of all polynomials p in P4 such that p(0) = 0 satisfies all three conditions, it is a subspace of P4.

(d) The set of all polynomials in P4 having at least one real root is not a subspace of P4.

To prove this, we only need

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Adeline had a box full of notebooks. If she gives 13 notebooks to each

Answers

Adeline has 60 notebooks and 4 children in the given case.

Let's assume that Adeline has x notebooks and y children.

According to the problem, if she gives 13 notebooks to each child, she will have 8 left. This can be expressed as:

x - 13y = 8 --- equation 1

Also, if she gives 15 notebooks to each child, she will have zero left. This can be expressed as:

x - 15y = 0 --- equation 2

We can solve these equations simultaneously to find the values of x and y.

Multiplying equation 1 by 15 and equation 2 by 13, we get:

15x - 195y = 120 --- equation 3

13x - 195y = 0 --- equation 4

Subtracting equation 4 from equation 3, we get:

2x = 120

x = 60

Substituting the value of x in equation 2, we get:

60 - 15y = 0

y = 4

Therefore, Adeline has 60 notebooks and 4 children.

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Adeline has a box full of notebooks. If she gives 13 notebooks to each child, she will have 8 left. If she gives 15 notebooks to each child, she will have zero left. How many notebooks and how many children does Adeline have?

complete solution pls
thankyou
Solve for the inverse Matrix 1. Using Adjoint of a Matrix A=2 -1 0
0 1 2
1 1 0 2.Using Gauss Jordan A=1 3
2 5
3. Using Equations and Identity Matrix A=1 -2
2 -3

Answers

The inverse of matrix A is:

A^-1 = |-3 -2|

|-2 -1|

Using Adjoint of a Matrix A=2 -1 0

0 1 2

1 1 0

The first step is to find the determinant of the matrix A:

|A| = 2(10 - 21) - (-1)(10 - 12) + 0(11 - 10)

= 4 + 2 + 0

= 6

Next, we need to find the adjoint of matrix A, which is the transpose of its cofactor matrix. The cofactor matrix is obtained by taking the determinant of the submatrix obtained by removing each element of the original matrix in turn and multiplying it by (-1)^(i+j), where i and j are the row and column indices of the removed element, respectively.

Cofactor matrix of A is

| 1 2 -1|

|-2 0 2|

|-1 2 1|

Taking the transpose of the cofactor matrix, we get the adjoint matrix of A as follows:

A^T = | 1 -2 -1 |

| 2 0 2 |

|-1 2 1 |

To find the inverse of A, we use the formula:

A^-1 = (1/|A|) A^T

Substituting the values, we get:

A^-1 = (1/6) | 1 -2 -1 |

| 2 0 2 |

|-1 2 1 |

Using Gauss Jordan A= |1 3|

|2 5|

We can find the inverse of a matrix using Gauss-Jordan elimination method as follows:

|1 3|1 0| |1 3|0 1|

|2 5|0 1|-> |0 1|-2/3 -1/3|

Therefore, the inverse of matrix A is:

A^-1 = |-2/3 -1/3|

| 1/3 1/3|

Using Equations and Identity Matrix A= |1 -2|

|2 -3|

We can find the inverse of a matrix A using the equations AX=I, where I is the identity matrix and X is the matrix that represents the inverse of A. The solution is given by:

|1 -2| |x11 x12| |1 0|

|2 -3| |x21 x22| = |0 1|

Multiplying the matrices, we get:

x11 = -3

x12 = -2

x21 = -2

x22 = -1

Therefore, the inverse of matrix A is:

A^-1 = |-3 -2|

|-2 -1|

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10. The point (-5. 7) is located on the terminal arm of angle A in standard position. a) Determine the primary trigonometric ratios for ZA. (k/2.A 12.0/2) b) Determine the primary trigonometric ratios for an B that has the same sine as ZA, but different signs for the other two primary trigonometric ratios. c) Use a calculator to determine the measures of ZA and 2B, to the nearest degree.

Answers

a) The primary trigonometric ratios for angle A in standard position are;

sin(A) = 7/√74, cos(A) = -5/√74, and tan(A) = -7/5.

b) The primary trigonometric ratios for angle B are;

sin(B) = 7/√74, cos(B) = -5/√74, and tan(B) = 7/5.

c) A ≈ -56° and 2B ≈ -69°

a) To find the primary trigonometric ratios (sine, cosine, tangent) for angle A in standard position, we need to use the coordinates of the point (-5, 7). We can find the hypotenuse by using the Pythagorean theorem:

h = √((-5)² + 7²)

h = √74

Then, we can use the definitions of sine, cosine, and tangent:

sin(A) = y/h = 7/√74

cos(A) = x/h = -5/√74

tan(A) = y/x = -7/5

So , the primary trigonometric ratios for angle A in standard position are;

sin(A) = 7/√74, cos(A) = -5/√74, and tan(A) = -7/5.

b) To find an angle B with the same sine as angle A but different signs for the other two primary trigonometric ratios, we can use the fact that;

⇒ sin(B) = sin(A).

We also know that the signs of cos(B) and tan(B) will be different from those of cos(A) and tan(A), since angle B will be in a different quadrant.

Since sin(B) = sin(A), we know that the y-coordinate of angle B will be the same as that of angle A, namely 7.

We can then use the Pythagorean theorem to find the x-coordinate:

x = √(h² - y²)

x = √(74 - 49)

x = √25

x = 5

Since angle B is in a different quadrant from angle A, we need to adjust the signs of cos(B) and tan(B) accordingly.

We know that cos(B) will be negative, since angle B is in the third quadrant where x is negative.

We also know that tan(B) will be positive, since angle B is in the second quadrant where y is positive and x is negative.

Therefore, we have:

cos(B) = -x/h = -5/√74

tan(B) = y/x = 7/5

So the primary trigonometric ratios for angle B are;

sin(B) = 7/√74, cos(B) = -5/√74, and tan(B) = 7/5.

c) To find the measure of angle A, we can use the inverse tangent function:

A = tan⁻¹ (-7/5)

A ≈ -56.31°

To find the measure of angle 2B, we can use the double angle formula for sine:

sin(2B) = 2sin(B)cos(B)

We already know sin(B) and cos(B) from part (b), so we can plug them in:

sin(2B) = 2(7/√74)(-5/√74)

sin (2B) = -70/37

We can then use the inverse sine function to find the measure of angle 2B:

2B = sin⁻¹(-70/37)

2B ≈ -68.59°

So, to the nearest degree, we have A ≈ -56° and 2B ≈ -69°.

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A line passes through the points (3,-2) and (3, 4). Determine the slope of the line.
0 m =
O m = 0
O The slope is undefined.
Om = 3

Answers

Since the line passes through the points (3,-2) and (3, 4), the slope of the line is: C. the slope is undefined.

How to calculate or determine the slope of a line?

In Mathematics and Geometry, the slope of any straight line can be determined by using the following mathematical equation;

Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Slope (m) = rise/run

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

By substituting the given data points into the formula for the slope of a line, we have the following;

Slope (m) = (4 + 2)/(3 - 3)

Slope (m) = (6)/(0)

Slope (m) = undefined.

Based on the graph, the slope is the change in y-axis with respect to the x-axis and it is undefined.

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I NEED ANSWERS
b=10m с a=7m ​

Answers

Answer:

c = 12.2

Step-by-step explanation:

a squared + b squared = c sqared

7 squared + 10 squared = 49+ 100 = 149 = [tex]\sqrt{x} 149[/tex] = 12.2

pls help asap will give points

Answers

The size of the angle X is calculated to be equal to 47.4° to the nearest tenth of degree using trigonometric ratios cosine

What is trigonometric ratios?

The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.

The basic trigonometric ratios includes;

sine, cosine and tangent.

Given the triangle STU;

cos X = ST/SU {adjacent/hypotenuse}

cos X = 8.8/13

X = cos⁻¹(8.8/13) {cross multiplication}

X = 47.3963°

Therefore, the measure of the angle X is calculated to be equal to 47.4° to the nearest tenth of degree using trigonometric ratios cosine

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A boat's heading is E15


N with a speed of 16 knots. The current is moving NW at 5 knots. What is the actual speed of the boat rounded to the nearest tenth?

Answers

The real speed of the vessel, adjusted to the closest tenth, is 20.4 hitches.

How to Solve the Problem?

To unravel this issue, we got to break down the boat's speed into its components. We will use trigonometry to discover the eastbound and northward components of the boat's speed.

First, let's draw a diagram:

              N

                |

                |

     NW 5 |   E15

                |

--------------|---------------> E

                |

                |

                |

               S

From the chart, we will see that the northward component of the boat's speed is:

16 hitches * sin(15°) = 4.16 knots

And the eastbound component of the boat's speed is:

16 ties * cos(15°) = 15.38 knots

Next, we ought to discover the whole northward and eastbound speed of the vessel by including the boat's speed components to the current's speed components. We are able moreover utilize trigonometry to discover the northward and eastbound components of the current's speed:

5 hitches * sin(45°) = 3.54 ties northward

5 hitches * cos(45°) = 3.54 ties eastbound

So, the overall northward speed of the pontoon is:

4.16 ties + 3.54 hitches = 7.7 hitches northward

And the full eastbound speed of the pontoon is:

15.38 ties + 3.54 ties = 18.9 ties eastbound

Presently, we will utilize the Pythagorean hypothesis to discover the greatness of the boat's speed vector:

sqrt((7.7 knots)^2 + (18.9 knots)^2) = 20.4 ties

In this manner, the real speed of the vessel, adjusted to the closest tenth, is 20.4 hitches.

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Thema deposits 500$ ina savings account with a simple interest rate of 1. 3%. How could you use this information to find the interest she would earn in 4 years and determine the percent change in her savings account

Answers

The percent change in her savings account is 5.2%, meaning her savings account has increased by 5.2% due to the interest earned over 4 years.

To find the interest Thema would earn in 4 years, we can use the simple interest formula:

where I is the interest earned, P is the principal (initial deposit), r is the interest rate (as a decimal), and t is the time period in years.

In this case, we have P = $500, r = 0.013, and t = 4. Plugging in these values, we get:

I = [tex]($500) (0.013) (4) = $26[/tex]

So, Thema would earn $26 in interest over 4 years.

To determine the percent change in her savings account, we need to compare the amount she will have after 4 years to the initial deposit. After 4 years, her savings account will have:

A = P + I = $500 + $26 = $526

The percent change in her savings account can be calculated as:

percent change = (new amount - old amount) / old amount x 100%

Substituting the values, we get:

percent change = ($526 - $500) / $500 x 100% = 5.2%

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