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There will be approximately 4621 fish in the lake after 7 months

To solve this problem, we can use the formula for exponential growth:

[tex]N = N0 * (1 + r)^t[/tex]

where N is the final population size, N0 is the initial population size, r is the monthly growth rate (in decimal form), and t is the number of m

onths.

In this case, the monthly growth rate is 6% or 0.06, and the monthly loss rate is 500 fish. So the net monthly growth rate is:

[tex]r_{net}[/tex] = 0.06 - 500/N0

Plugging in the given values, we have:

[tex]r_{net}[/tex]= 0.06 - 500/3500

= 0.0457

Now we can use the formula above to find the population size after 7 months:

[tex]N = 3500 * (1 + 0.0457)^7[/tex]

= 4621.42

So the final population size after 7 months, rounded to the nearest whole number, is:

N ≈ 4621

Therefore, there will be approximately 4621 fish in the lake after 7 months, taking into account both the monthly growth rate and the monthly loss rate.

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a)An equation of / is y = 2x + 1.

b) The exact length of AC is 2√17.

c) The exact length of BM is √17.

d) The coordinates of B are (5, 4).

(a) Since / is the perpendicular bisector of AC, it passes through the midpoint M of AC. The coordinates of M can be found by taking the average of the x-coordinates and the average of the y-coordinates of A and C, respectively:

x-coordinate of M = (-3 + 5)/2 = 1

y-coordinate of M = (4 + 2)/2 = 3

Therefore, the coordinates of M are (1, 3). Since / is perpendicular to AC, its slope is the negative reciprocal of the slope of AC:

slope of AC = (2 - 4)/(5 - (-3)) = -1/2

slope of / = 2 (negative reciprocal of -1/2)

Using the point-slope form of the equation of a line with the point M, we get:

y - 3 = 2(x - 1)

Simplifying and rearranging, we get:

y = 2x + 1

Therefore, an equation of / is y = 2x + 1.

(b) The length of AC can be found using the distance formula:

AC = √[(5 - (-3))^2 + (2 - 4)^2] = √64 + 4 = √68 = 2√17

Therefore, the exact length of AC is 2√17.

(c) Since BM is a median of triangle ABC, it divides AC into two equal parts. Therefore, BM has length half of AC:

BM = AC/2 = (1/2)(2√17) = √17

Therefore, the exact length of BM is √17.

(d) Since B lies on line /, its x-coordinate is 5 (since / passes through point (5, 2)). To find the y-coordinate, we can substitute x = 5 into the equation of /:

y = 2x + 1 = 2(5) + 1 = 11

Therefore, the coordinates of B are (5, 11).

Alternatively, we can use the fact that B lies on the circumcircle of triangle ABC (since angle ABC is a right angle) to find its coordinates. The circumcenter of triangle ABC is the midpoint of AC (which is also the intersection of the perpendicular bisectors of AC), so the coordinates of the circumcenter are (1, 3). The radius of the circumcircle is half of AC, so it is √17. Therefore, the equation of the circumcircle is:

(x - 1)^2 + (y - 3)^2 = 17

Since B lies on the circumcircle, its coordinates satisfy this equation. Substituting x = 5, we get:

(5 - 1)^2 + (y - 3)^2 = 17

16 + (y - 3)^2 = 17

(y - 3)^2 = 1

y - 3 = ±1

Solving for y, we get y = 2 or y = 4. Since B lies on line /, its y-coordinate must be greater than 2, so y = 4. Therefore, the coordinates of B are (5, 4).

(e) There are two possible positions of B, which are reflections of each other across the line /. One position has coordinates (5, 11), as found in part (d), and the other has coordinates (-3, 2). To find the coordinates of the second position, we can reflect point A across line / to get the point B':

B' has the same x-coordinate as A, which is -3. To find its y-coordinate, we can use the equation

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The cash payment equivalent to making monthly payments of $1074 for 5 years with an interest rate of 11% per annum compounded semi-annually is $49,701.

To find the cash payment, we can use the Present Value of the Annuity formula:

PV = PMT * [(1 - (1 + r)⁻ⁿ) / r]

Where:
PV = Present Value (cash payment)
PMT = Monthly payment ($1074)
r = Monthly interest rate
n = Number of payments (5 years * 12 months = 60)

First, we need to find the monthly interest rate. Since the interest is compounded semi-annually, we'll divide the annual interest rate by 2, and then convert it to a monthly rate:

Semi-annual interest rate = 11% / 2 = 5.5% = 0.055
[tex](1 + 0.055)^{(1/6)} - 1[/tex] ≈ 0.008944 (rounded to 6 decimal places)

Now we can plug the values into the Present Value of Annuity formula:

PV = 1074 * [(1 - (1 + 0.008944)⁻⁶⁰) / 0.008944]
PV ≈ 1074 * [1 - (0.5861) / 0.008944]
PV ≈ 1074 * 46.2768
PV ≈ 49,701

The cash payment equivalent is approximately $49,701 (rounded to the nearest dollar).

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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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Answer:  16.8 feet

Note: your teacher may not want you to enter "feet" and instead may just want the number only.

===================================================

Work Shown:

tan(angle) = opposite/adjacent

tan(40) = h/20

20*tan(40) = h

h = 20*tan(40)

h = 16.7819926 which is approximate

h = 16.8

The streetlamp is approximately 16.8 ft tall.

When using your calculator, make sure it's in degree mode. One way to check is to compute something like tan(45) and you should get 1 as a result.

Step-by-step explanation:

Answer:

63 mph

Step-by-step explanation:

10.5 × 6 is 63

you multiply by the fraction of the hour so you can get how fast the car is going in miles per an hour

Answer:

The average speed of the car = 63miles per hour

Step-by-step explanation:

Given, the total distance traveled by the car= 10.5 miles

total time is taken by the car to cover 10.5 miles = 1/6 hour

formula to calculate average speed

average speed = total distance/total time

average speed = 10.5/(1/6)

                          = 10.5×6

                          = 63 miles per hour

therefore, the average speed of the car is 63 miles per hour

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

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:

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.

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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Answer:

h = -1.25

Step-by-step explanation:

isolate the variable by dividing each side by factors that don't contain the variable.

If the price level in Canada falls and the price level in the United States does not change, Canadian-manufactured sofas are relatively less expensive, so Sherri will likely purchase a Canadian manufactured sofa.

This is because the decrease in price level in Canada would make Canadian goods more affordable, including Canadian manufactured sofas. This makes them a more attractive option for Sherri than U.S. manufactured sofas which would still be relatively more expensive even if their price level did not change. Answer D is correct in this scenario. However, if U.S. manufactured sofas were initially more expensive than Canadian sofas, then answer B could also be correct. Overall, Sherri's decision would depend on the initial price difference between Canadian and U.S. manufactured sofas, as well as her personal preferences and any other relevant factors.

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The surface area of the entire prism is 294 ft².

The surface area of the entire prism can be found by summing the areas of the triangular and rectangular faces of the prism.

Since we have two triangular faces and  3 rectangular faces. Thus,

surface area of the entire prism = 2*( 1/2bh) + 3*(L*W)

where b = 6, h = 4, L = 18 and W = 5

surface area of the entire prism = 2*( 1/2 * 6*4) + 3*(18*5)

surface area of the entire prism = 24  + 270

surface area of the entire prism = 294 ft²

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

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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[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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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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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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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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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]

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

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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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.

The probability P(56< X ≤ 6) is 0.0398. ​To draw a normal curve with the area corresponding to the probability shaded we would shade the area to the right of 56 and to the left of 6 on the curve.

To compute the probability P(56< X ≤ 6), we first need to standardize the values using the formula:
z = (x - μ) / σ
where x is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.

For 56:
z = (56 - 45) / 10 = 1.1

For 6:
z = (6 - 45) / 10 = -3.9

Now, we can look up the probabilities for these values of z in a standard normal distribution table or use a calculator to find the area under the curve between these two values:
P(56< X ≤ 6) = P(1.1 < Z ≤ -3.9)

Using a standard normal distribution table or a calculator, we find that:
P(56< X ≤ 6) = 0.0398 (rounded to four decimal places)

To draw the normal curve with the shaded area corresponding to this probability, we can use a graphing calculator or a standard normal distribution table. The area between 56 and 6 corresponds to the area to the right of 56 minus the area to the right of 6. So, we would shade the area to the right of 56 and to the left of 6 on the curve, which would look like this:
[insert normal curve with shaded area between 56 and 6]

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The statement that describe the situation are;

E. While on the first roller coaster, the height switches from positive to negative approximately every 40 seconds,

D.While on the second roller coaster, the height switches every 80 seconds.

Since Function is a type of relation, or rule, that maps one input to specific single output. It is Linear function which is a function whose graph is a straight line

While on the first roller coaster, we can see that the function modeling Michael's height switches from positive to negative approximately every 40 seconds, meaning he changes from being at a height above the platform to below the platform approximately every 40 seconds.

While on the second roller coaster, we can see that this change occurs every 80 seconds.

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Answer:

I beleive c. i had the same question last year, and i believe i got the answer right. so sorry if its wrong. hope this helps.

Step-by-step explanation:

The relation R defined as R = {(x,x) : 1 € Z} on Z, where Z is the set of integers, is a relation where an element in Z is related to itself if and only if it is equal to 1.

To determine whether the relation R is reflexive, symmetric, and transitive, we need to consider the properties of relations.

A relation is reflexive if every element in the set is related to itself. In this case, since R contains only pairs of the form (x,x), we can say that R is reflexive if and only if 1 € Z. That is, if and only if 1 is an integer, then R is reflexive. Since 1 is an integer, R is reflexive.

A relation is symmetric if for any two elements (a, b) in the relation, (b, a) is also in the relation. Since R only contains pairs of the form (x,x), it is symmetric if and only if for any integer x, (x,x) is in the relation, then (x,x) is also in the relation. Therefore, R is symmetric.

A relation is transitive if for any three elements (a, b), (b, c) in the relation, (a, c) is also in the relation. In this case, since R only contains pairs of the form (x,x), we can say that R is transitive if and only if for any integers x, y, z such that (x, y) and (y, z) are in R, then (x, z) is also in R. However, since there are no pairs (x, y) and (y, z) in R except for when x=y=z=1, there are no pairs (x, z) in R for which transitivity needs to be checked. Therefore, we can say that R is transitive vacuously.

In conclusion, the relation R defined as R = {(x,x) : 1 € Z} on Z is reflexive, symmetric, and transitive.

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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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The missing angle outside the triangle is 137 degrees.

The missing angle in the triangle can be found as follows:

Vertically opposite angles are congruent.

Therefore,

180 - 45  -  60  = (sum of angles in a triangle)

180 - 105 = 75 degrees

Therefore,

180 - 75 - 68 = 37 degrees

Using the exterior angle theorem,

The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles of the triangle.

Hence,

let

x = missing angle

100 + 37 = x

x = 137 degrees

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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?

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

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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The surface area of the tissue box is 240 cm².

Option B is the correct answer.

We have,

The tissue box can be considered a triangular prism.

The formula for the surface area of the tissue box can be made as:

The perimeter of the triangular base x height of the box.

Now,

Perimeter

= 8 + 6 + 10

= 24 cm

And,

The surface area of the tissue box.

= 24 x 10

= 240 cm²

Thus,

The surface area of the tissue box is240 cm².

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The distance from y to W is sqrt(10), which is the exact answer using radicals.

Let's start by finding the projection of y onto the subspace W spanned by v1 and v2. The projection of y onto W is given by:

projW(y) = ((y · v1)/||v1||^2)v1 + ((y · v2)/||v2||^2)v2

where · denotes the dot product and || || denotes the norm or length of a vector.

Using the given information, we have:

v1 = [1 0 1 0], v2 = [0 1 0 1], and y = [2 4 0 0]

We can calculate the dot products and norms as follows:

||v1||^2 = 1^2 + 0^2 + 1^2 + 0^2 = 2

||v2||^2 = 0^2 + 1^2 + 0^2 + 1^2 = 2

y · v1 = 2(1) + 4(0) + 0(1) + 0(0) = 2

y · v2 = 2(0) + 4(1) + 0(0) + 0(1) = 4

Therefore, the projection of y onto W is:

projW(y) = ((2/2)[1 0 1 0]) + ((4/2)[0 1 0 1])

= [1 0 1 0] + [0 2 0 2]

= [1 2 1 2]

The closest point to y in W is the projection projW(y), so we have:

v = [1 2 1 2]

The distance from y to W is the length of the vector y - v, which we can calculate as:

||y - v|| = ||[2 4 0 0] - [1 2 1 2]||

= ||[1 2 -1 -2]||

= sqrt(1^2 + 2^2 + (-1)^2 + (-2)^2)

= sqrt(10)

Therefore, the distance from y to W is sqrt(10), which is the exact answer using radicals.

Complete question: Let [tex]$y=\left[\begin{array}{r}13 \\ -1 \\ 1 \\ 2\end{array}\right], y_1=\left[\begin{array}{r}1 \\ 1 \\ -1 \\ -2\end{array}\right]$[/tex], and [tex]$v_2=\left[\begin{array}{l}5 \\ 1 \\ 0 \\ 3\end{array}\right]$[/tex] . Find the distance from y to the subspace W of [tex]$\mathrm{R}^4$[/tex] spanned by [tex]$v_1$[/tex] and [tex]$v_2$[/tex], given that the closest point to [tex]$y$[/tex] in [tex]$W$[/tex] is [tex]$\hat{y}=\left[\begin{array}{r}11 \\ 3 \\ -1 \\ 4\end{array}\right]$[/tex].

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