Answer:
Let's start by translating the sentence into an equation using algebra.
Let's say the number is represented by "x".
According to the problem:
"Twice a number" can be written as "2x".
"If twice a number is decreased by 13" can be written as "2x - 13".
"The result is 9" can be written as "= 9".
Putting it all together, we get the equation:
2x - 13 = 9
Now we can solve for x:
2x = 22
x = 11
Therefore, the number is 11.
Choose the correct item from each drop-down menu to factor the trinomial 3x2 + 13x – 10 by grouping
The factored form of 3x^2 + 13x - 10 is (3x - 2)(x + 5).
To factor 3x^2 + 13x – 10 by grouping, we need to find two numbers whose product is 3(-10) = -30 and whose sum is 13.
Let's list all the factor pairs of -30:
-1, 30
-2, 15
-3, 10
-5, 6
Out of these pairs, the pair that adds up to 13 is -2 and 15.
We can use these numbers to rewrite the middle term:
3x^2 - 2x + 15x - 10
Now, we can group the first two terms and the last two terms:
(3x^2 - 2x) + (15x - 10)
We can factor out the greatest common factor from each group:
x(3x - 2) + 5(3x - 2)
Now, we can see that we have a common factor of (3x - 2):
(3x - 2)(x + 5)
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The given question is incomplete, the complete question is:
What is the factored form of 3x^2 + 13x - 10?
Every year, Johnathan the lumberjack cuts down all the trees on his farm. He finds that %60 of the trees die, but the rest grow again from the stumps that remain. What percent of the original trees are still alive after 3 years?
After answering the presented question, we can conclude that As a equation result, approximately 44.4% of the original trees are still living after three years.
What is equation?An equation is a mathematical statement that proves the equality of two expressions connected by the equal symbol '='. 2x - 5 Equals 13, for example. Expressions include 2x-5 and 13. The character '=' joins the two expressions. A mathematical formula with two algebraic expressions on either side of an equal sign (=) is known as an equation. It demonstrates the relationship of equivalence between the left and right formulas. In every formula, LHS = RHS (left side = right side).
Assuming that all of the trees on the farm are cut down each year, the percentage of trees that survive after three years is as follows:
60% of the trees die after the first year, leaving only 40% of the original trees.
Following the first year, 40% of the original trees are still standing.
Following the second year, 40% of the original trees remain + (40% of the original trees * 100% - 60% = 40% stump growth) = 64% of the original trees remain.
Following the third year, 40% of the original trees remain + (64% of the original trees * 100% - 60% = 4.4% stump growth) = 44.4% of the original trees remain.
As a result, approximately 44.4% of the original trees are still living after three years.
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What percent of data population falls between a z-score of -1. 5 and 0. 5
Around 62.47% of the data population is situated between a z-score of -1.5 and 0.5.
Assuming a standard normal distribution, the percentage of the data population that falls between a z-score of -1.5 and 0.5 can be calculated using a standard normal table or a calculator.
Using a standard normal table, we can find the area under the curve between z = -1.5 and z = 0.5.
The area to the left of z = 0.5 is 0.6915, and the area to the left of z = -1.5 is 0.0668. Therefore, the area between z = -1.5 and z = 0.5 is:
0.6915 - 0.0668 = 0.6247
So, approximately 62.47% of the data population falls between a z-score of -1.5 and 0.5.
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The length of Anna's square pencil case is 6 in and the width is 4 1/2 in what is the area of the pencil case?
If the length of Anna's square pencil case is 6 in and the width is 4 1/2 in, then the area of the pencil case is 27 square inches
The area of a rectangle can be calculated by multiplying its length by its width.
In this case, Anna's pencil case is a square, which means that its length and width are equal. The length of the pencil case is given as 6 inches, so its width is also 6 inches.
Therefore, the area of the pencil case is:
Area = length x width
Substitute the values in the equation
Area = 6 in x 4.5 in
Area = 27 square inches
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Geometry: Find the unknown lengths of these 30-60 right triangles (ASAP!!!)
The lengths of the sides of the special 30°–60° right triangles are;
14. a = 8
b = 8·√3
c = 16
15. a = 4
b = 4·√3
c = 8
16. a = 7
b = 7·√3
c = 14
17. a = 12
b = 12·√3
c = 24
18. a = 5
b = 5·√3
c = 10
19. a = 9
b = 9·√3
c = 18
20. a = 10
b = 10·√3
c = 20
21. a = (√6)/2
b = 3·(√(2))/s
c = √6
What are special right triangles?Special right triangles are triangles that have 30°, 60°, and 45° interior angles.
The unknown lengths of the 30-60 special right triangle can be found as follows;
14. The length of the side a = 8
The length of the hypotenuse side, c can be found as follows;
sin(30°) = 1/2
sin(θ) = a/c
sin(30°) = 1/2 = 8/c
c = 2 × 8 = 16
c = 16
cos(θ) = b/c
cos(30°) = √3/2
Therefore; √3/2 = b/16
b = 16 × (√3/2) = 8·√3
b = 8·√3
15. a = 4, therefore;
c = 2 × 4 = 8
c = 8
b = (√3/2) × 8 = 4·√3
b = 4·√3
16. a = 7
b = 7·√3
c = 2 × 7 = 14
c = 14
17. a = b/√3, therefore;
a = 12·√3/√3 = 12
a = 12
b = 12·√3
c = 2 × a, therefore;
c = 2 × 12 = 24
c = 24
18. a = 5
b = a × √3, therefore;
b = 5 × √3 = 5·√3
b = 5·√3
c = 2 × a, therefore;
c = 2 × 5 = 10
c = 10
19. c = 18
a = c/2, therefore;
a = 18/2 = 9
a = 9
b = a × √3, therefore;
b = 9 × √3 = 9·√3
b = 9·√3
20. a = 10
b = a × √3, therefore;
b = 10 × √3 = 10·√3
b = 10·√3
c = 2 × a, therefore;
c = 2 × 10 = 20
c = 20
21. c = √6
a = c/2, therefore;
a = (√6 )/2
b = a × √3, therefore;
b = ((√6)/2) × √3 = (√(18))/2 = 3·√2/2
b = 3·√2/2
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Can someone pls answer this
How do these numbers compare?
Drag the correct comparison symbol to the box.
10.09 10.9
< = >
Addressing issue at hand, we can state that the correct linear equation comparison symbol to the box. 10.09 10.9 f < = > is => 10.09 < 10.9
What is a linear equation?The algebraic equation y=mx+b is known as a linear equation. M serves as the y-intercept, and B serves as the slope. The previous clause has two variables, y and x, and is sometimes referred to as a "linear equation with two variables". Bivariate linear equations are those with two independent variables. The following are a few examples of linear equations: 2x - 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, and 3x - y + z = 3. When an equation takes the form y=mx+b, with m denoting the slope and b denoting the y-intercept, it is referred to as being linear. The term "linear" refers to an equation having the form y=mx+b, where m stands for the slope and b for the y-intercept.
the correct comparison symbol to the box.
10.09 10.9
< = >
10.09 < 10.9
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how to algebraically find all the real zeros on a polynomial function
Answer:
To find all the real zeros of a polynomial function algebraically, use the Rational Root Theorem to generate a list of possible rational zeros and use synthetic division to test each possible zero. The zeros that remain after testing all the possible zeros are the real zeros of the polynomial function.
To find all the real zeros of a polynomial function algebraically, you can use the Rational Root Theorem and synthetic division. The Rational Root Theorem states that if a polynomial function has integer coefficients, then any rational zero (i.e., a zero that can be expressed as a fraction) must have a numerator that is a factor of the constant term and a denominator that is a factor of the leading coefficient.
Here are the steps you can follow:
1. Write the polynomial function in descending order of degree, with all like terms combined.
2. Use the Rational Root Theorem to generate a list of possible rational zeros. This list will be a set of all the possible fractions that can be formed by dividing a factor of the constant term by a factor of the leading coefficient.
3. Use synthetic division to test each possible zero. Start with the smallest possible denominator and work your way up. If a possible zero is not a zero of the polynomial function, cross it off the list.
4. Repeat step 3 until all the possible zeros have been tested. The zeros that remain are the real zeros of the polynomial function.
For example, let's say we have the polynomial function f(x) = x^3 - 6x^2 + 11x - 6.
Write the polynomial function in descending order of degree: f(x) = x^3 - 6x^2 + 11x - 6.
Use the Rational Root Theorem to generate a list of possible rational zeros: ±1, ±2, ±3, ±6.
Use synthetic division to test each possible zero. We start with x = 1:
1 │ 1 -6 11 -6
│ 1 -5 6
└─────────────
1 -5 6 0
Since the remainder is zero, we have found a zero of the polynomial function at x = 1. We can write the factorization f(x) = (x - 1)(x^2 - 5x + 6).
Now we use synthetic division to test the other possible zeros:
2 │ 1 -6 11 -6
│ 2 12 46
└─────────────
1 -4 23 40
x = 2 is not a zero of the polynomial function.
3 │ 1 -6 11 -6
│ 3 15 78
└─────────────
1 -3 26 72
x = 3 is not a zero of the polynomial function.
-1 │ 1 -6 11 -6
│ -1 7 -4
└────────────
1 -7 18 -10
x = -1 is not a zero of the polynomial function.
-2 │ 1 -6 11 -6
│ -2 16 -50
└────────────
1 -8 27 -56
x = -2 is not a zero of the polynomial function.
-3 │ 1 -6 11 -6
│ -3 27 -102
└────────────
1 -9 38 -108
x = -3 is not a zero of the polynomial function.
6 │ 1 -6 11 -6
Hope this helps, I'm sorry if it doesn't! :]
madison started a bank account with $200. each year, she earns 5% in interest, which means the amount of money in the account is multiplied by 1.05 each year.if madison does not withdraw money from her account, how much will she have in 10 years?
Madison will have $322.65 in her bank account in 10 years. Here is the calculation to find this amount:
Starting balance: $200
Year 1: $200 x 1.05 = $210
Year 2: $210 x 1.05 = $220.50
Year 3: $220.50 x 1.05 = $231.53
...
Year 10: $279.10 x 1.05 = $322.65
Therefore, after 10 years, Madison will have $322.65 in her bank account due to the annual interest earned.
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can yall help me pls???????
Answer:45%
Step-by-step explanation:
:))))
Answer:
45%
Step-by-step explanation:
99/180 = 55%
100% - 55% = 45% decrease
Answer: 45%
A cone radius 10 cm and height 18cm fits exactly over a cylinder so that the cylinder is just touching hte inside surface of the cone. The radius of the cylinder is 4cm,
Pls help me I only have 1 day lefhelp pls! ITS Prediction from line of best Fit
Answer:79
Step-by-step explanation:
y=12+91=79
i need help on these questions
The value of x which would make GHJK a rhombus would be 42 degrees.
Nora can construct a kite with diagonals that bisect each other if the kite is in the shape of a rhombus.
How to find the value of x ?The way that a Rhombus is drawn is such that ∠ GKJ and ∠ HGK would add up to 180 degrees. If these angles are bisected as shown, then the corresponding angles would add up to 90 degrees as a result.
If this shape is to be a rhombus then the value of x would be :
= 90 - 48
= 42 degrees
What kind of kite has diagonals that bisect ?A kite is a quadrilateral with two pairs of adjacent congruent sides. The diagonals of a kite intersect at a right angle, but they do not necessarily bisect each other.
If the diagonals of a kite bisect each other, it means that they cut each other into two equal parts. This can only happen if the kite is a rhombus, which is a special type of kite where all four sides are congruent.
Therefore, if Nora wants to construct a kite with diagonals that bisect each other, she would need to ensure that the kite is a rhombus.
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help pleaseeeeeeeeeeeeeeeeeeeeeeeeeee
Answer:
yea
Step-by-step explanation:
u need more details
Find the quotient of 40y^(4)-5y^(3)-30y^(2)-10y divided by 5y
The quotient of [tex]40y^(4)-5y^(3)-30y^(2)-10y[/tex] divided by [tex]8y^(3) - y^(2) - 6y - 2[/tex]
You can use the polynomial long division method to find the product of two polynomials. Following are the steps:
Step 1: Descending in degree, write the dividend polynomial. If any terms are absent from the polynomial, replace them with coefficients of 0.Step 2: Descend the degree of the divisor polynomial as you write it. If any terms are missing from the divisor polynomial, substitute coefficients of 0 for those terms.Step 3: To calculate the first term of the quotient, divide the dividend's first term by the divisor's first term.Step 4: Multiply the divisor by the first term of the quotient to get the first term of the partial product.Step 5: Subtract the partial product from the dividend.Step 6: Bring down the next term of the dividend.Step 7: Repeat steps 3-6 until all the terms of the dividend have been used.Step 8: The final result is the quotient, plus any remainder left over after the last subtraction.To divide [tex]40y^(4)-5y^(3)-30y^(2)-10y[/tex] by 5y, we utilize long division. Divide dividend by divisor to obtain first term of the quotient:
[tex]40y^(4) / (5y) = 8y^(3)[/tex]
Multiply divisor (5y) by first term of quotient (8y^(3)) to obtain first term of partial product:
[tex](8y^(3))(5y) = 40y^(4)[/tex]
Subtract the partial product from the dividend to obtain remainder:
[tex]40y^(4) - 40y^(4) = 0[/tex]
[tex]-5y^(3) / (5y) = -y^(2)(-y^(2))(5y) = -5y^(3)-5y^(3) - (-5y^(3)) = 0[/tex]
[tex]-30y^(2) / (5y) = -6y(-6y)(5y) = -30y^(2)-30y^(2) - (-30y^(2)) = 0[/tex]
Repeat process:
[tex]-10y / (5y) = -2(-2)(5y) = -10y-10y - (-10y) = 0[/tex]
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How many days can the baby drink the juice if no one else drinks more
The baby will drink 45 liters of milk in 30 days when baby drink 3 liters of milk in 2 days.
To solve this problem, we can set up a proportion using the fact that the amount of milk the baby drinks is directly proportional to the number of days:
milk consumed / number of days = constant
Let's call the constant of proportionality k. Then we can write:
3 liters / 2 days = k
To find the value of k, we can solve for it:
k = 3 liters / 2 days = 1.5 liters/day
Now we can use this value of k to find how much milk the baby will drink in 30 days:
milk consumed / 30 days = 1.5 liters/day
Multiplying both sides by 30, we get:
milk consumed = 1.5 liters/day x 30 days = 45 liters
Therefore, the baby will drink 45 liters of milk in 30 days.
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The complete question is :
A baby drink 3 liters of milk in 2 days . How much milk will baby drink in 30 days.
A coffee pot holds 2 quarts of coffee. How much is this in cups?
A coffee pot that holds 2 quarts is equal to 8 cups. To determine the number of cups in a coffee pot that holds 2 quarts, multiply 2 quarts by 4 cups.
A quart is a unit of measurement that is equal to 4 cups of liquid. This means that a coffee pot that holds 2 quarts is equal to 8 cups. To figure out how many cups are in a quart, use the following equation: 1 quart = 4 cups. To determine the number of cups in a coffee pot that holds 2 quarts, multiply 2 quarts by 4 cups. The result is 8 cups. In other words, a coffee pot that holds 2 quarts is equal to 8 cups of liquid.
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Sam invests $500 for two years at an interest rate of 12%, compounded twice a year. How much will his total investment be worth after 2 years?
Answer:
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal (initial amount)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the time the money is invested (in years)
In this case, P = $500, r = 0.12, n = 2 (compounded twice a year), and t = 2.
So, A = 500(1 + 0.12/2)^(2*2) = $673.01
Therefore, Sam's total investment will be worth $673.01 after 2 years.
Step-by-step explanation:
A tissue box is shaped like a rectangular prism the tissue box measures 5. 2cm wide 9. 6cm long and 6 cm tall approximately what is the volume of the tissue box
the approximate volume of the tissue box to the nearest whole number is 300 cubic centimeters.
The volume of a rectangular prism is given by the formula V = lwh, where l, w, and h are the length, width, and height of the prism, respectively.
Substituting the given values, we get:
V = (5.2 cm)(9.6 cm)(6 cm)
Simplifying, we get:
V = 299.52 cubic centimeters
Therefore, the approximate volume of the tissue box is 299.52 cubic centimeters.
Since the dimensions of the tissue box are given to only two decimal places, we can reasonably assume that the answer accurate to two decimal places is a good approximation. If we want to express the answer to the nearest whole number, we would round it to the nearest integer.
Rounding 299.52 to the nearest integer, we get:
V ≈ 300 cubic centimeters
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The length of human pregnancies is approximately normal with mean μ=266days and standard deviation σ=16days.
(a) What is the probability that a randomly selected pregnancy lasts less than 262 days?
(b) Suppose a random sample of 51 pregnancies is obtained. Describe the sampling distribution of the sample mean length of pregnancies.
(c) What is the probability that a random sample of 51 pregnancies has a mean gestation period of 262 days or less?
(d) What is the probability that a random sample of 106 pregnancies has a mean gestation period of 262 days or less?
The probability that a random sample of 106 pregnancies has a mean gestation period of 262 days or less is 0.00003.
What is probability?
Probability is a measure of the likelihood that an event will occur. It is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. Probabilities are usually expressed as fractions, decimals, or percentages.
(a) To find the probability that a randomly selected pregnancy lasts less than 262 days, we need to standardize the value using the formula z = (x - μ) / σ, where x is the value we're interested in, μ is the mean, and σ is the standard deviation.
z = (262 - 266) / 16 = -0.25
Using a standard normal distribution table or calculator, we find that the probability of getting a z-score of -0.25 or less is 0.4013. Therefore, the probability that a randomly selected pregnancy lasts less than 262 days is 0.4013.
(b) The sampling distribution of the sample mean length of pregnancies is approximately normal with mean μ = 266 and standard deviation σ = 16 / sqrt(51) = 2.2449. This is known as the central limit theorem.
Using a standard normal distribution table or calculator, we find that the probability of getting a z-score of -2.8304 or less is 0.0023. Therefore, the probability that a random sample of 51 pregnancies has a mean gestation period of 262 days or less is 0.0023.
(d) Using the same formula as in part (c), we have:
z = (262 - 266) / (16 / sqrt(106)) = -4.0077
Using a standard normal distribution table or calculator, we find that the probability of getting a z-score of -4.0077 or less is 0.00003.
Therefore, the probability that a random sample of 106 pregnancies has a mean gestation period of 262 days or less is 0.00003.
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Who no the answer to this
Answer: 5/6
Step-by-step explanation:
First, you find the common denominator of the two:
3/6 + 2/6
Now you can easily add the two:
5/6
Answer:
See below.
Step-by-step explanation:
We are asked to Add Fractions.
What are Fractions?
Fractions are parts of a single whole number.
To make adding these fractions similar, we should have a common denominator.
[tex]\frac{Numerator}{Denominator}[/tex]
When we add common denominators, they remain the same. This makes adding fractions much easier.
Let's identify the common denominator by finding the Least Common Multiple. (LCM)
What is the LCM?
The LCM is the smallest common multiple that 2 numbers have. A multiple is the product of that number.
An Example of Multiples:
[tex]4; 4 \times 1, 4 \times 2, 4 \times 3, 4 \times 4, 4 \times 5, 4 \times 6.\\4; 4, 8, 12, 16, 20, 24.[/tex]
An Example of the LCM:
[tex]4; 4, 8, 12, 16, 20, [24], 28\\6; 6, 12, 18, [24], 30, 36, 42. \\The \ LCM \ is \ 24.[/tex]
Let's use the example above as a guide, but for our values in the denominator.
[tex]2; 2, 4, [6], 8, 10, 12, 14.\\3; 3, [6], 9, 12, 15, 18, 21.\\The \ LCM \ is \ 6.[/tex]
Our Denominators will be 6.
Multiply both the Numerator and the Denominator by the number needed to make the denominator 6.
[tex]\frac{1 \times 3}{2 \times 3}=\frac{3}{6} \\\frac{1 \times 2}{3 \times 2} = \frac{2}{6}[/tex]
Add:
[tex]\frac{3}{6} + \frac{2}{6} = \frac{5}{6} \\[/tex]
Our final answer is [tex]\frac{5}{6} .[/tex]
how do i solve 7/9 +8 1/4 fraction
Answer:
9 1/36
Step-by-step explanation:
set 7/9 and 1/4 to a common denominator
so lets do 36,
7 x 4 = 28
9 x 4 = 36
1 x 9 =9
4 x 9 =36
28/36 + 9/36 = 1 1/36
1+8 = 9
The angle between the lines of sight from a lighthouse to a tugboat and to a cargo ship is 27°. The angle between the lines of sight at the cargo ship is twice the angle between the lines of sight at the tugboat. What are the angles at the tugboat and at the cargo ship?
The angle between the line of sight from the lighthouse to the tugboat is 27° and the angle between the line of sight from the lighthouse to the cargo ship is 54°.
Let's call the angle between the line of sight from the lighthouse to the tugboat "x". Then, we know that the angle between the line of sight from the lighthouse to the cargo ship is x+27, since the given angle between the lines of sight is 27°.
We also know that the angle between the lines of sight at the cargo ship is twice the angle between the lines of sight at the tugboat. Using this information, we can set up the equation:
2x = x+27
Solving for x, we get:
x = 27
So the angle between the line of sight from the lighthouse to the tugboat is 27°. Then, we can use this to find the angle between the line of sight from the lighthouse to the cargo ship:
x+27 = 27+27 = 54
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PLEASE HELPP I NEED THISS
Kiana rides her skateboard with a constant speed of 6 km/h. How long will she take to travel a distance of 10 kilometers?
Answer:
100 minutes
Step-by-step explanation:
we take 6/km an hour and make it 1 km per 10 minutes and then multiply it by 100
192192 is what percent of 600600600?192192 is what percent of 600600600?
Answer:
0.032%
Step-by-step explanation:
192192/600600600 x 100% = 0.00032 x 100%
0.032%
Please help me I tried I got a 60
Answer:
X = 90 , Y = 37 and Z = 53 Degrees.
Step-by-step explanation:
Z = 53 as opposite angles of a parallelogram are equal
Therefore, Z + Y + 90 = 180 ( Angle sum Property of a triangle)
= 53 + Y + 90 = 180
Y = 180 - 143
Y = 37
Now to find x , we know Y is the Alternate Interior angle of the angle adjacent to the 90*
Keeping that angle as "o"
Angle O = Angle Y by Alternate Interior Angle Therom
Therefore , Angle X + Angle O + 53 = 180 (ASP of triangle)
Angle X + 37 + 53 = 180
Angle X = 180 - 90
Angle X = 90*
In summary , X = 90 , Y = 37 and Z = 53.
Hope it helps.
7.03 The Unit Circle
The concept of the unit circle is presented throughout it's answer.
What is the unit circle?The unit circle is a circle with a radius of 1 unit that is centered at the origin (0,0) of a coordinate plane. It is a fundamental concept in trigonometry and geometry, and it is used to define the trigonometric functions (sine, cosine, and tangent) of angles in the Cartesian plane.
The format of each coordinate on the unit circle is given as follows:
[tex](\cos{\theta}, \sin{\theta})[/tex].
Hence we can calculate the trigonometric measures for any angle, as the tangent, the secant, the cossecant and the cotangent of an angle are all functions of the sine and the cosine obtained by the unit circle.
Missing InformationThe problem is incomplete, hence the unit circle concept is presented in this answer.
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identify the area of the figure
Answer: 184 cm^2
Step-by-step explanation:
1st: separate the figure into smaller shapes
-> one large rectangle (20 cm x 7 cm) =140 cm
-> one triangle ( (15-7)= 1/2 (8 cm x 3 cm ) = (20-13-4) = 12 cm
-> one rectangle (4 cm x 8 cm ) = 32 cm
140 + 32 + 12 = 184 cm
Answer: 184 cm2
Step-by-step explanation: Break the shape into common shapes. If you continue the line of 13 cm you now have a rectangle and trapezoid.
The rectangle area is 140
20 x 7 = 140
The trapezoid area is 44
The 15 cm line is cut into 7 and 8
the height of the shape is now 8
Base A ( the top ) is 4 cm
Base B (the bottom) is 7
A = a+b over 2h = 4+7 over 2·8 = 44
now add the numbers
HELP!
A rectangular fish tank has a width w inches, length w+ 8 inches, and height 18 - w inches. All dimensions are greater than 6 inches. The volume of the tank is 1440 cubic inches. How many inches is the height of the fish tank?
As per the volume, the height of the fish tank is 0.4 inches.
First, we are given that the width of the tank is w inches, the length is w+8 inches, and the height is 18-w inches. We also know that all dimensions are greater than 6 inches. Using the formula for the volume of a rectangular prism, we can write:
V = lwh
Substituting the given values, we get:
1440 = (w+8)(w)(18-w)
Now, we can simplify this equation by expanding the product on the right-hand side:
1440 = 18w² + 8w(18-w)
Simplifying further, we get:
1440 = 18w² + 144w - 8w²
Combining like terms, we get:
10w² + 144w - 1440 = 0
Dividing both sides by 10, we get:
w² + 14.4w - 144 = 0
Now we can solve for w using the quadratic formula:
w = (-b ± √(b² - 4ac)) / 2a
where a = 1, b = 14.4, and c = -144. Plugging in these values, we get:
w = (-14.4 ± √(14.4² - 4(1)(-144))) / 2(1)
Simplifying, we get:
w = (-14.4 ± √(432.16)) / 2
w = (-14.4 ± 20.8) / 2
w = -17.6 or w = 3.6
Since we know that all dimensions are greater than 6 inches, we can eliminate the negative solution and conclude that w = 3.6 inches is not valid. Therefore, the width of the tank is w = 17.6 inches.
Now we can use this value to find the height of the tank. Substituting w = 17.6 into the expression for the height, we get:
h = 18 - w
h = 18 - 17.6
h = 0.4
Therefore, the height of the tank is h = 0.4 inches, or equivalently, 4/10 of an inch.
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Determine the equation of the tangent line to the given path at the specified value of t. (Enter your answer as a comma-separated list of equations in (x, y, z) coordinates.) (sin(3t), cos(3t), 2t^9/2)); t = 1
The equation of the tangent line to the given path at the specified value of t is (x - sin(3), 3cos(3)(y - cos(3)), -3sin(3)(z - 1), 9/2).
To determine the equation of the tangent line to the given path at the specified value of t, we need to find the derivative of the given path with respect to t. The derivative of the given path is (3cos(3t), -3sin(3t), 9t^8/2). Now, we can plug in the specified value of t = 1 to find the slope of the tangent line at that point. The slope of the tangent line at t = 1 is (3cos(3), -3sin(3), 9/2).
Next, we can find the point on the given path at t = 1 by plugging in the specified value of t into the original equation. The point on the given path at t = 1 is (sin(3), cos(3), 1).
Finally, we can use the point-slope form of an equation to find the equation of the tangent line. The equation of the tangent line in (x, y, z) coordinates is (x - sin(3)) = 3cos(3)(y - cos(3)) = -3sin(3)(z - 1) = 9/2.
Therefore, the equation of the tangent line to the given path at the specified value of t is (x - sin(3), 3cos(3)(y - cos(3)), -3sin(3)(z - 1), 9/2).
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Find the volume generated by rotating the region in the first quadrant bounded by y =e" and the X-axis from = 0 to x = ln(3) about the y-axis. Express your answer in exact form. Volume =
The volume generated by rotating the region in the first quadrant bounded by y = ex and the x-axis from x = 0 to x = ln(3) about the y-axis is (πln(3)3)/3.
To find the volume generated by rotating the region in the first quadrant bounded by y = ex and the x-axis from x = 0 to x = ln(3) about the y-axis, we can use the disk method. The disk method involves slicing the region into thin disks and adding up their volumes.
The volume of each disk is πr2h, where r is the radius of the disk and h is the thickness of the disk. In this case, the radius of each disk is x and the thickness is dx.
So, the volume of the region is:
V = ∫0ln(3)πx2dx
We can use the power rule for integration to solve this integral:
V = π∫0ln(3)x2dx = π[(x3)/3]0ln(3) = π[(ln(3)3)/3 - (03)/3] = (πln(3)3)/3
Therefore, the volume generated by rotating the region in the first quadrant bounded by y = ex and the x-axis from x = 0 to x = ln(3) about the y-axis is (πln(3)3)/3. This is the exact form of the volume.
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