In the figure, the triangles are similar. What is the distance d from the water ride to the roller coaster? Express your answer as a decimal rounded to the nearest tenth if necessary.

Answer:

701.122

Step-by-step explanation:

Based on the question on rational functions;

1. The domain in interval notation is: (-∞, -7) ∪ (-7, 1) ∪ (1, 8) ∪ (8, ∞)

2. The discontinuities are:

x = 1: Hole (since (x-1) was canceled out)

x = -7: Vertical asymptote (since it's still in the denominator)

x = 8: Vertical asymptote (since it's still in the denominator)

3. If a factor in the denominator can be canceled out with a factor in the numerator, then the discontinuity at that value of x is a hole. If a factor in the denominator cannot be canceled out, then the discontinuity at that value of x is a vertical asymptote.

4.  The end-behavior of r(x) is determined by the ratio of the highest-degree terms, which is x/x² = 1/x. Using limit notation, we have:

lim (x → ∞) r(x) = 0

lim (x → -∞) r(x) = 0

5.  Since the degree of the numerator (1) is less than the degree of the denominator (2), the horizontal asymptote is y = 0.

6. The x-intercept is (-6, 0). The y-intercept is (0, - 0.12)

For the rational function, to determine whether each discontinuity is a hole or a vertical asymptote, we factor the numerator and denominator to see if there is any common factor that can be canceled out.

(x) = [(x-1)(x+6)] / [(x-1)(x+7)(x-8)]

We can see that (x-1) is a common factor and can be canceled out. So, the simplified function is:

r(x) = (x+6) / [(x+7)(x-8)]

To find the x-intercepts, set r(x) equal to 0 and solve for x:

0 = (x+6) / [(x+7)(x-8)]

This means x + 6 = 0, so x = -6. The x-intercept is (-6, 0).

To find the y-intercept, set x = 0:

r(0) = (0+6) / [(0+7)(0-8)] = 6 / (-56) = -3 / 28

So, the y-intercept is (0, -3/28).

The above answers is based on the following questions as seen in the picture;

Rational Functions

1. Consider the following rational function

r(x) = (x2-x)(x+6)

        (x-1)(x+7)(x-8)

a. Write the domain of r(x)in interval notation

b. Find the discontinuities of r(x) and label each as a hole or a vertical asymptote.

c. Explain how you decide if there is a hole or a vertical asymptote. d. What terms in r(x)determine the end-behavior? Describe the end-behavior of r(x), using limit notation.

e. If there is a horizontal asymptote, write the equation of the

horizontal asymptote

f. Find the x- and y-intercepts of r(x)

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Answer:

C. 1,620 m³

Step-by-step explanation:

always remember, for a volume we need to (one way or another) multiply measurements of 3 dimensions with each other.

so, if each of the 3 dimension measurements triples, then that factor 3 is used in the multiplication 3 times.

and the volume is multiplied by 3×3×3 = 27.

and 60 m³ × 27 = 1,620 m³

The answers to all parts is shown below.

We have,

An oceanographer claims that the mean dive duration of a North Atlantic right whale is minutes.

a) Now, the hypothesis are

[tex]H_0[/tex] : [tex]\mu[/tex] = 11.3

and, [tex]H_\alpha[/tex] : [tex]\mu[/tex] ≠ 11.3

So, the claim is Null Hypothesis

b) t= (x- [tex]\mu_0[/tex])/ (s/ √n)

t= (12.2-11.3)/ (2.2/ √34)

t= 2.39

P value = 0.023

c) Reject [tex]H_0[/tex] because the p value is not greater than α.

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Answer:

The tan(a) is the degree value of the angle shown. In this case, the unknown degree is A, and it wants you to use the Tangent method. Using the formula to get 27°

Step-by-step explanation:

Tangent (A) = opposite (√17)  over adjacent (8)

So it would be Tan(A)= √17/8

To easily solve this use a calculator

Find the tan button, (use the inverse tangent or tan-1)

then sub values

tan-1(√17/8)

27.26 or 27°

For future questions here's a helpful chart:

The 20th and the 75th percentiles of the P/E ratios, can be found to be :

To find the 20th percentile:

Arrange the data in ascending order:

17, 18, 18, 19, 19, 20, 22, 22, 22, 22, 24, 25, 29, 31, 34, 38

Calculate the index of the 20th percentile:

Index = (20/100) x n = (20/100) x 16 = 3.2

Since 3.2 is not a whole number, we need to round up to the next integer to get the index of the smallest value that is greater than or equal to the 20th percentile. Therefore, the index is 4.

The 20th percentile is the value at the 4th index, which is 19.

To find the 75th percentile:

Index = (75/100) x n = (75/100) x 16 = 12

The 75th percentile is the value at the 12th index, which is 31.

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By algebra properties, the product of the two binomials (4 · x + 1) · (2 · x² - 2) is equal to (4 · x) · (2 · x²) + (- 2) · (4 · x) + 1 · (2 · x²) + 1 · (- 2). (Correct choice: B)

In this question we find the product of two binomials, the former is a linear binomial and the latter is a quadratic binomial. This can be done by algebra properties. First, write the entire expression:

(4 · x + 1) · (2 · x² - 2)

Second, use distributive property twice:

(4 · x + 1) · (2 · x²) + (4 · x + 1) · (- 2)

(4 · x) · (2 · x²) + 1 · (2 · x²) + (- 2) · (4 · x) + (- 2) · 1

8 · x³ + 2 · x² - 8 · x - 2

Third, use commutative property:

(4 · x) · (2 · x²) + (- 2) · (4 · x) + 1 · (2 · x²) + 1 · (- 2)

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a negative slope would best fit the data.  [tex]y=-1x+7.5[/tex]Thus, option C is correct.

We must discover the equation of the straight line that goes through the given data points and best captures their general trend in order to identify the best linear equation that fits the data points.

To begin, we can plot the points on a graph and examine the shape that they take.

The fact that the data points appear to be moving downward from left to right suggests that a linear equation with a negative slope would best fit the data.

By decreasing the sum of the squared angles between each point and the line that best fits the points.

which is a step in the least squares regression technique, we may ascertain the slope of the line that best fits the data.The equation for the best-fit line, which is, [tex]y = -0.675x + 6.35[/tex]

There aren't any available answers for this equation. We can see, however, that while answer choice C comes the closest to having the right slope (-1), the y-intercept is incorrect.

Therefore, the answer closest to the best-fit equation is: C. [tex]y=-1x+7.5[/tex]

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Answer:

Step-by-step explanation:

To solve this problem, we need to find the slope of the line that represents the increase in sales over the years. We can use the formula for the slope of a line:

slope = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two points on the line. We can choose the points (3, 165000) and (8, 320000) since we know the sales in their third and eighth business years.

slope = (320000 - 165000) / (8 - 3)

slope = 31000

This means that for each additional business year, the sales increase by $31,000. To predict the sales in their tenth business year, we can use the point-slope form of the equation of a line:

y - y1 = m(x - x1)

where m is the slope we just found, and (x1, y1) is any point on the line. We can use (8, 320000) as the point, since we know the sales in their eighth business year and want to predict the sales in their tenth business year, which is two years later.

y - 320000 = 31000(x - 8)

Simplifying the equation, we get:

y = 31000x + 64000

To find the predicted sales in their tenth business year (x = 10), we substitute x = 10 into the equation:

y = 31000(10) + 64000

y = 310000 + 64000

y = 374000

Therefore, the predicted sales in their tenth business year is $374,000.

Answer:   $2,800

Step-by-step explanation:

Since each computer is worth $400, we know that for 3 computers, the price would be $1,200 (400 x 3 = 1,200). We know that the fourth computer would have a discount of 50% off.  50% is equal to 1/2, so 400 x 1/2 is equal to $200. For four computers, the price would be $1,400. If we want to find the amount for 8 computers, we have to do 1,400 x 8. The answer to this is 2,800.

the number of cars that entered at IN is [tex]12[/tex].

Based on these observations, we can set up an equation to solve for the number of cars entering at IN:

Let's assume the number of cars entering at IN is "x".

Number of cars exiting at C [tex]= 12[/tex] (given)

Number of cars exiting at  [tex]A = 0[/tex] (since cars exiting at A do not return)

Number of cars exiting at   [tex]B = 0[/tex] (since cars exiting at B do not return)

Number of cars exiting at C = Number of cars entering at C + Number of cars exiting at A + Number of cars exiting at B

[tex]12 = x + 0 + 0[/tex]

Simplifying the equation, we get:

[tex]12 = x[/tex]

Therefore, the number of cars that entered at IN is 12.

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Let's assign the variable "x" to the height of the smaller plant. Since we know that the larger plant grows twice as fast, its height would be "2x". Let's say the total height of the plants together is 90 cm, so we can write the equation: x + 2x = 90

Let's assign the variable "x" to the height of the smaller plant. Since we know that the larger plant grows twice as fast, its height would be "2x". Let's say the total height of the plants together is 90 cm, so we can write the equation:

x + 2x = 90

Simplifying the equation, we get:

3x = 90

x = 30

Therefore, the height of the smaller plant is 30 cm and the height of the larger plant is 2x = 60 cm.

To sketch a graph of the equation, we can plot the values of x (height of smaller plant) on the x-axis and the values of 3x (total height of the two plants) on the y-axis. The graph would be a straight line with a slope of 3 and intersecting the y-axis at 90 (the total height of the plants).

For the second equation, let's say we want to model the growth of the smaller plant over time. We can use the equation:

y = 0.5x + 10

where "y" is the height of the smaller plant (in cm) and "x" is the number of weeks since planting. The equation assumes that the plant starts at a height of 10 cm and grows at a rate of 0.5 cm per week. To sketch the graph, we can plot the values of x on the x-axis and the values of y on the y-axis. The graph would be a straight line with a slope of 0.5 and intersecting the y-axis at 10.

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The value of f'(0) in given equation is -1/3.

This is the case because quadratum is the Latin word for square, and since the area of a square of side length x is given by x², a polynomial equation having exponent two is known as a quadratic ("square-like") equation.

Since the integral of f(x) is a rational function, we can use the reverse power rule to find the derivative of f(x). According to the reverse power rule, if

[tex]∫f(x)x^n dx[/tex] = F(x) + C

where F(x) is a rational function, then

f(x) = d/dx (F(x) + C) = F'(x)

Now, let's evaluate the given integral:

∫[tex]f(x)x^2(x-7)^7[/tex]dx

Using integration by substitution with u = x-7 and du = dx, we can rewrite the integral as:

[tex]∫f(u+7)(u+7)^2 u^7[/tex]du

Expanding (u+7)², we get:

∫[tex]f(u+7)(u^2+14u+49) u^7[/tex]du

Expanding further and distributing f(u+7), we get:

∫[tex](f(u)u^9 + 14f(u)u^8 + 49f(u)u^7 + ... + f(u)7^9 + 14f(u)7^8 + 49f(u)7^7)[/tex]du

Since the integral is a rational function, the only non-zero term in the expansion must be a constant multiple of u². Therefore, we have:

∫[tex]f(x)x^2(x-7)^7[/tex] dx = k ∫u^2 du = (1/3)ku³ + C

where k is some constant.

Now, we know that f(0) = -2, so we can find k as follows:

∫[tex]f(x)x^2(x-7)^7[/tex] dx = (1/3)k(0-7)³ + C = -343k/3 + C

But we also know that the integral is a rational function, so it must be equal to (ax² + bx + c)/d for some constants a, b, c, and d. Therefore,

(1/3)k(0-7)³ + C = (ax² + bx + c)/d

Substituting x = 0 and using f(0) = -2, we get:

(1/3)k(-7)³ + C = c/d = -2

Solving for k, we have:

(1/3)k(-7)³+ C = -2

k = -18/(343C)

Substituting this expression for k into the integral, we have:

∫[tex]f(x)x^2(x-7)^7[/tex]dx = (-6/(49C))u³+ C'

where C' is some constant. Finally, using the reverse power rule, we have:

f(x) = d/dx (-6/(49C))(x-7)³ + C''

f(0) = -2 implies C'' = -2

So, at x = 0, we have:

f'(0) = (-6/(49C)) * 3(-7)² = 18/(49C)

Substituting the expression we derived for k, we get:

f'(0) = 18/(49(-18/(343C))) = -1/3

Therefore, the value of f'(0) is -1/3.

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The expected value of a prize is $245.00.

In probability theory, a probability distribution is a function that describes the likelihood of obtaining the different possible outcomes of a random variable. The function assigns a probability to each possible outcome, such that the probabilities add up to 1.

There are two types of probability distributions: discrete and continuous. A discrete probability distribution is used when the random variable takes on a finite or countable number of values, such as the outcome of rolling a dice or the number of students in a class. A continuous probability distribution is used when the random variable can take on any value within a certain range, such as the height or weight of a person.

Now the expected value of a prize is the sum of each possible prize value multiplied by its corresponding probability.

Using the table provided, we can calculate the expected value as:

Expected value = $0.00(0.81) + $100.00(0.1) + $500.00(0.07) + $10,000.00(0.02)

Expected value = $0.00 + $10.00 + $35.00 + $200.00

Expected value = $245.00

Therefore, the expected value of a prize is $245.00.

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The solution to the quadratic equation is;

x = -2 ± √27/5

We can see that the equation that we should solve by the method of completing the square is given by;

5x^2 + 20x – 7 = 0

Dividing through by 5 we have;

x^2 + 4x - 7/5 = 0

Thus;

x^2 + 4x = 7/5

When we add 2^2 to both sides to complete the square we have;

(x + 2)^2 = 7/5 + 4

(x + 2)^2 =7 + 20/5

(x + 2)^2 =27/5

x = -2 ± √27/5

This is how the equation that we have can be solved by completing the square method.

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The completion of the table on simple interest would be :

The simple interest will be the same every year which is why it would keep increasing by $ 30 for each year up to the 4 years. It will go from $ 30 to $ 60 to $ 90, and then $120 so that the total in the account after 4 years is:

= 600 + 120

= $ 720

The simple interest formula to help find out after 4 years is:

= Principal + ( Principal x Rate x Time)

= 600 + ( 600 x 5 % x 4 )

= 600 + 120

= $ 720

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5.

y = -x + 5

6.  the point of contact is (-1, 6).

7.  the radius is √26

8. The standard form equation of a circle with center (h, k) and radius r is: (x - 2)^2 + (y - 3)^2 = 26

9. Line AB = AB

Line BC = BC

Line AC = AC

10. The equation of a circle with center (h, k) and radius r is:

(x - 4)^2 + (y + 3)^2 = 16^2

the equation of the line passing through (2, 3) with slope -1:

y - 3 = (-1)(x - 2)

y - 3 = -x + 2

y = -x + 5

We can find the intersection point of this line and the given tangent line by solving the system of equations:

y = x + 7

y = -x + 5

When we add these equations, we have:

2y = 12

y = 6

We can then substitute into either equation gives x = -1.

Hence, the point of contact is (-1, 6).

We  find the radius, Using the distance formula, :

r = √[(2 - (-1))^2 + (3 - 6)^2] = √26

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Answer:

The customer has 3 choices for dessert: brownie, cookie, or cake.

For bread, the customer can choose from 3 options: wheat, rye, or white.

Therefore, the customer has 3 choices for dessert and 3 choices for bread, making a total of 3 x 3 = 9 different meal choices.

The large count condition is met because all expected counts are greater than 5.

Now to determine if the data provides convincing evidence that the two population proportions differ, we can conduct a hypothesis test.

Let p1 be the proportion of adults who exercise regularly and got sick in the past year, and p2 be the proportion of adults who do not exercise regularly and got sick in the past year.

The null hypothesis is that the two population proportions are equal: H0: p1 = p2

The alternative hypothesis is that the proportion of those who exercise regularly who got sick in the past year is smaller than the proportion of those who do not exercise regularly: Ha: p1 < p2

We can use a two-sample z-test for proportions to test this hypothesis. The test statistic is given by:

where p_hat is the pooled sample proportion:

p_hat = (x1 + x2) / (n1 + n2)

In this case, we have:

n1 = 50, x1 = 18, n2 = 75, x2 = 56

p_hat = (18 + 56) / (50 + 75) = 0.455

We can also calculate the expected counts using the formula provided:

n1p_hat = 500.455 = 22.75

n1*(1-p_hat) = 500.545 = 27.25

n2p_hat = 750.455 = 34.13

n2(1-p_hat) = 75*0.545 = 40.88

The large counts condition is met because all expected counts are greater than 5.

To calculate the test statistic, we have:

z = (0.18 - 0.747) / sqrt(0.592 * (1 - 0.592) * (1/50 + 1/75)) = -2.36

Using a standard normal table, we find the p-value to be 0.0091 for a one-tailed test. Since the p-value is less than 0.05, we reject the null hypothesis and conclude that there is convincing evidence that the proportion of those who exercise regularly who got sick in the past year is smaller than the proportion of those who do not exercise regularly.

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A system of equations that can be used to determine the dollar amount of phone sales is:

P = 2C

0.07P + 0.03C = 85

Let's define the variables:

Let P represent the total dollar amount of phone sales Colton made.

Let C represent the total dollar amount of computer sales Colton made.

We are given that Colton earns a 7% commission on phone sales and a 3% commission on computer sales. We are also given that Colton had twice as much in phone sales as he had in computer sales and earned a total of $85 in commission.

We can write the following system of equations:

P = 2C (Colton had twice as much in phone sales as he had in computer sales)

0.07P + 0.03C = 85 (Colton earned a total of $85 in commission)

Now we have a system of equations:

P = 2C

0.07P + 0.03C = 85

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The p-value of the test statistic is given as follows:

0.0606.

The z-score of a measure X of a normally distributed variable that has mean represented by [tex]\mu[/tex] and standard deviation represented by [tex]\sigma[/tex] is obtained by the equation presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

We have a right-tailed test, as we are testing if the proportion is greater than a value.

As we have a right-tailed test, the p-value is one subtracted by the p-value of z = 1.55, hence:

1 - 0.9394 = 0.0606

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Answer:

Hello, The answer to your question is,

[tex]x=[/tex] −[tex]\frac{2}{t^3}[/tex]

hope this help :)
(correct me if i am wrong and im sorry)

Step-by-step explanation:

Answer:

2.5 years. Since the interest compounded is quarterly and an annual interest rate of 10% (That's $100 dollars per interest compounded), this means that after a year you will have gained $400 dollars in interest. Thus in order to double you money ($2,000) it will take 2 and a half years.

Step-by-step explanation:

Double your money:  2 x $1,000 = $2,000 (This will be the final total)

10% of 1000 is 100.

Quarterly = 4

4 x $100 = $400 back in interest per year.

2.5 x $400 = $1,000

or

$1,000/$400 = 2.5 (years)

$1,000 (from the initial deposit) + $1,000 (from interest) = $2,000

Answer:

Step-by-step explanation:

True.

To calculate Jane's debt-to-income ratio, we need to divide her total liabilities by her total income. However, the prompt only gives us the sum of Jane's assets and liabilities, not her income.

Assuming that Jane's assets are not considered as part of her income, we can conclude that she falls within the recommended amount, since her total liabilities ($2,300) divided by her total income (which we don't know but is presumably greater than her liabilities) would be less than 15%.

The average speed of the canoe toward the lighthouse was approximately 12.8 feet per minute.

Let's denote the distance between the canoe and the lighthouse as "x" and the height of the lighthouse as "h".

Then, using trigonometry, we can set up the following equations:

tan(6.5°) = h / (x + 115)

tan(48.7°) = h / (x + 115)

Solving for "h" in each equation and setting them equal to each other, we get:

h / (x + 115) = h / (x + 115)

tan(6.5°) / (x + 115) = tan(48.7°) / (x + 115 - 5)

Simplifying and solving for "x", we get:

x = 64.1 feet

Therefore, the canoe traveled a distance of 64.1 feet in 5 minutes.

To find the average speed, we can divide the distance by the time:

average speed = 64.1 feet / 5 minutes = 12.8 feet per minute

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The value in coins of 30 galleons is 1250 coins

From the question, we have the following parameters that can be used in our computation:

25 coins = 2 doubloons

3 galleons = 10 doubloons

Multiply the second equation by 10

So, we have

30 galleons = 100 doubloons

Multiply the first equation by 50

So, we have

1250 coins = 10 doubloons

So, we have

30 galleons = 100 doubloons = 1250 coins

This means that

30 galleons = 1250 coins

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Step-by-step explanation:

to find the area you first have to seperate the shape into the triangle on the bottom and the trapezoid on the top. to find the area of the triangle its

[tex]3 \times 2 - 2 \times 2 \div 2 - 1 \times 2 \div 2 = 6 - 2 - 0.5 = 6 - 2.5 = {3.5cm}^{2} [/tex]

then to find the area of the trapezoid its

[tex](4 + 6) \div 2 \times 2 = 10 \div 2 \times 2 = 5 \times 2 = {10cm}^{2} [/tex]

then add both of them together and its 13.5 cm squared.

A graph of the lattice points of the inverse are shown below.

In Mathematics, an inverse function simply refers to a type of function that is obtained by reversing the mathematical operation in a given function (f(x)).

In this scenario and exercise, you are required to determine the inverse of the function f(x). This ultimately implies that, we would have to swap (interchange) both the independent value (x-value) and dependent value (y-value) as follows;

f(x) = y = log₂(-x + 3) - 4

x = log₂(-y + 3) - 4

log₂(-y + 3) = x + 4

[tex]2^{x + 4} = -y + 3\\\\y = -2^{x + 4} + 3[/tex]

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Complete Question:

The function f(x) = log₂(-x + 3) - 4 is graphed below. Plot all lattice points of the inverse. Use the labeled points as your guide.

Answer:

85 square inches.

Step-by-step explanation:

The area of a triangle can be found by the formula [tex]1/2bh[/tex], assuming b is the base and h is the height.

If b is 17 inches and h is 10 inches:

[tex]1/2(17)(10)[/tex]

[tex]1/2(170)[/tex]

[tex]A = 85[/tex]

Therefore, the area is 85 square inches.