Determine whether Rolle's theorem applies to the function shown below on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle's theorem. f(x)=x(x−3)2,[0,3] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. Rolle's Theorem applies and the point(s) guaranteed to exist is/are x= (Type an exact answer, using radicals as needed Use a comma to separate answers as needed ) B. Rolle's Theorem does not apply

Answer:

Alex has 3x dollars and an extra 15 dollars in his coat pocket. He buys a new Nike shoe for 84 dollars. How much money did Alex spend that was not in his pocket.

Step-by-step explanation:

Answer:

Part A: Word problem

Maria went to the store and purchased some books for her book club. Each book cost $3, and she also bought some bookmarks at $15 each. Maria's total purchase, including tax, amounted to $84. If Maria bought x books, write an equation to represent the situation.

Part B: Solution

To solve the equation 3x + 15 = 84, we need to isolate the variable x on one side of the equation.

Step 1: Subtract 15 from both sides of the equation to eliminate the constant term on the left side:

3x + 15 - 15 = 84 - 15

3x = 69

Step 2: Divide both sides of the equation by 3 to isolate x:

3x/3 = 69/3

x = 23

So, the solution to the equation is x = 23.

Part C: Explanation

In the given equation 3x + 15 = 84, the variable x represents the number of books Maria purchased. The equation states that the cost of x books at $3 each, represented by 3x, plus the cost of $15 for bookmarks, totals to $84. Thus, the value of x represents the number of books Maria bought in this scenario. In the solution, x = 23, it means Maria purchased 23 books for her book club.

Step-by-step explanation:

The graph of each transformed function should be matched to the verbal description as follows;

In Mathematics and Geometry, the translation a geometric figure or graph downward simply means adding a digit to the value on the y-coordinate of the pre-image or function.

In Mathematics, a vertical translation to the positive y-direction (downward) is modeled by this mathematical equation g(x) = f(x) - N.

Where:

By critically observing the graph represented by green A, we can reasonably infer and logically deduce that the graph of the parent function was reflected over the x-axis, and then followed by a horizontal translation to the left by 2 units;

f(x) = |x|

g(x) = -|x + 2|

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A probability generating function is a powerful tool for analyzing the properties of a geometrically distributed random variable. In particular, we can use it to determine the expected value and variance of T. The probability generating function for T is given by G(z) = p/(1-qz), where q = 1-p.

To find the expected value, we differentiate the generating function with respect to z and evaluate it at z=1. This yields E(T) = G'(1) = q/p. To find the variance, we differentiate the generating function twice with respect to z and evaluate it at z=1. This yields Var(T) = G''(1) + G'(1) - [G'(1)]^2 = (2-p)/p²
Thus, using the probability generating function, we have found that the expected value of T is q/p and the variance of T is (2-p)/p². These results are useful for understanding the behavior of T in various applications.

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The first term is 1, and the 17th term is 32,768.

A geometric series is a series in which each term is obtained by multiplying the preceding term by a constant factor called the common ratio (r). The general formula for a geometric series is:

a, ar, ar², ar³, ..., arⁿ⁻¹

where a is the first term, r is the common ratio, and n is the number of terms in the series.

Now, let's consider the given geometric series whose term is given by a₀ .rⁿ. We are given that a5 = 32 and a8 = 256. Using the general formula for a geometric series, we can write:

a, ar, ar², ar³, ar⁴, ar⁵, ...

where a = a₀, rⁿ = a5/a₀ = 32/a₀, and r⁸ = a8/a₀ = 256/a₀.

To find the value of r, we can divide the equation r⁸ = 256/a₀ by the equation r⁵ = 32/a₀, which gives:

(r⁸)/(r⁵) = (256/a₀)/(32/a₀) r³ = 8 r = 2

Therefore, the common ratio of the given geometric series is 2.

To find the value of a₀, we can substitute r = 2 and a5 = 32 in the equation rⁿ = 32/a₀ to get:

2ⁿ = 32/a₀ a₀ = 32/2ⁿ

Substituting n = 5, we get a₀ = 1.

Finally, to find the value of a17, we can use the formula for the nth term of a geometric series:

aₙ = a₀ . rⁿ⁻¹

Substituting a₀ = 1 and r = 2, we get:

a₁₇ = 1 . 2¹⁷⁻¹ = 32,768

Therefore, the value of a₁₇ in the given geometric series is 32,768.

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The derivative of the function g(x) = 3/x^5 + 2/x^3 + 6. 3√xg'(x) =  -45√x/x^8 - 18√x/x^6

To find the derivative of the function g(x) = 3/x^5 + 2/x^3 + 6, we use the power rule and the sum rule of differentiation:

g'(x) = -15/x^6 - 6/x^4

Now, we can simplify the expression for 3√xg'(x) by factoring out a common factor of 3/x^4:

3√xg'(x) = 3√x (-15/x^6 - 6/x^4)

Simplifying further, we can combine the two terms inside the parentheses by finding a common denominator:

3√xg'(x) = 3√x (-15/x^6 - 6/x^4) = 3√x (-15x^2 - 6x^4)/x^10

Simplifying the numerator, we get:

3√xg'(x) = -45√x/x^8 - 18√x/x^6

Therefore, 3√xg'(x) = -45√x/x^8 - 18√x/x^6.

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The probability that a randomly chosen arrival takes more than 12 minutes is approximately 0.0498 or 4.98%.

To solve this problem, we can use the fact that the time between arrivals in an exponential distribution follows the exponential distribution with parameter λ, where λ is the rate of arrivals per unit time.

In this case, the rate of arrivals is 15 patients per hour, or λ = 15/60 = 0.25 patients per minute.

Let X be the time between arrivals, then X follows an exponential distribution with parameter λ = 0.25.

To find the probability that a randomly chosen arrival takes more than 12 minutes, we need to calculate:

P(X > 12)

We can use the cumulative distribution function (CDF) of the exponential distribution to calculate this probability. The CDF of the exponential distribution is given by:

[tex]F(x) = 1 - e^(-λx)[/tex]

So, we have:

P(X > 12) = 1 - P(X ≤ 12)

= 1 - F(12)

= [tex]1 - (1 - e^(-0.25*12))[/tex]

=[tex]e^(-3)[/tex]

Therefore, the probability that a randomly chosen arrival takes more than 12 minutes is approximately 0.0498 or 4.98%.

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The value of the test statistic for testing H0: rho = 0 vs. H1: rho ≠ 0, given a correlation coefficient of 0.47 for a sample of 17 students, is approximately 2.06.

To find the value of the test statistic for testing H0: rho = 0 vs. H1: rho ≠ 0, given a correlation coefficient of 0.47 for a sample of 17 students, you can follow these steps:

Step 1: Calculate the degrees of freedom.
Degrees of freedom (df) = n - 2, where n is the sample size.
df = 17 - 2 = 15

Step 2: Use the formula for the test statistic, t:
t = (r * sqrt(df)) / sqrt(1 - r^2), where r is the correlation coefficient.
t = (0.47 * sqrt(15)) / sqrt(1 - 0.47^2)

Step 3: Calculate the value of the test statistic:
t = (0.47 * sqrt(15)) / sqrt(1 - 0.2209)
t = (0.47 * 3.87298) / sqrt(0.7791)
t = 1.8202046 / 0.88270386

t ≈ 2.06

Thus, The value of the test statistic is approximately 2.06.

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As per the concept of probability, there is approximately a 30.4% chance of selecting 2 or 3 defective drugs out of 10 selected at random for checking.

To solve this problem, we first need to find the probability of selecting a defective drug from the company's output. Since we are given that 3% of the output is defective, the probability of selecting a defective drug is 0.03.

Next, we need to use this probability to find the probability of selecting exactly 2 or 3 defective drugs out of 10. We can use the binomial probability formula for this:

P(X = x) = (n choose x) * pˣ * (1-p)ⁿ⁻ˣ

where P(X = x) is the probability of selecting k defective drugs out of n, p is the probability of selecting a defective drug, and (n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n items.

To find the probability of selecting exactly 2 or 3 defective drugs, we need to calculate P(X = 2) + P(X = 3). Plugging in the values, we get:

P(X = 2) = (10 choose 2) * 0.03² * 0.97⁸ ≈ 0.225

P(X = 3) = (10 choose 3) * 0.03³ * 0.97⁷ ≈ 0.079

Therefore, the probability of selecting 2 or 3 defective drugs out of 10 is:

P(X = 2 or X = 3) = P(X = 2) + P(X = 3) ≈ 0.225 + 0.079 ≈ 0.304 or 30.4%

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The exact value of the integral is approximately 18.184, while the trapezoidal rule approximation with n=5 is approximately 18.178.

To apply the trapezoidal rule, we need to divide the interval [6,10] into n=5 subintervals of equal width:

Δx = (10-6)/5 = 1.6

The endpoints of these subintervals are:

x0 = 6

x1 = 6 + Δx = 7.6

x2 = 6 + 2Δx = 9.2

x3 = 6 + 3Δx = 10.8

x4 = 6 + 4Δx = 12.4

The trapezoidal rule states that:

[tex]\int _a^b f(x) dx \approx \Delta x/2 [f(a) + 2f(x1) + 2f(x2) + ... + 2f(x(n-1)) + f(b)][/tex]

Applying this formula with a=6, b=10 and n=5, we have:

[tex]10\int 6^{10} 3/(x-4) dx \approx x/2 [f(6) + 2f(7.6) + 2f(9.2) + 2f(10.8) + f(12.4)][/tex]

where f(x) = 3/(x - 4)

f(6) = 3/(6-4) = 1.5

f(7.6) = 3/(7.6-4) = 0.7299

f(9.2) = 3/(9.2-4) = 0.5

f(10.8) = 3/(10.8-4) = 0.375

f(12.4) = 3/(12.4-4) = 0.2909

Substituting these values, we get:

[tex]10\int 6^{ 10} 3/(x-4) dx \approx 0.8 [1.5 + 2(0.7299) + 2(0.5) + 2(0.375) + 0.2909][/tex]

[tex]10\int 6^{10} 3/(x-4) dx \approx 18.178[/tex]

To find the exact value of the integral, we can use the antiderivative of f(x):

∫ 3/(x-4) dx = 3 ln|x-4| + C

where C is the constant of integration.

Using this formula, we have:

[tex]10\int 6^{ 10} 3/(x-4) dx = [10(3 ln|x-4|)]_ 6^10[/tex]

= 30 ln|10-4| - 30 ln|6-4|

= 30 ln(3) - 30 ln(2)

≈ 18.184.

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Answer: the value of the given integral is 5376/49.

Step-by-step explanation:

To evaluate the given integral by changing the coordinates, we need to determine the new region of integration in the uv-plane that corresponds to the parallelogram R in the xy-plane.

First, we solve the equations of the lines that bound the parallelogram R:

x - 2y = 0 --> y = (1/2)x

x - 2y = 4 --> y = (1/2)x - 2

3x - y = 1 --> y = 3x - 1

3x - y = 8 --> y = 3x - 8

Next, we make the change of variables u = x - 2y and v = 3x - y.

So, we have x = (2u + v)/7 and y = (v - u)/7.

Now, we need to express the original integral in terms of the new variables u and v.

The Jacobian of the transformation is:

J = ∂(x,y) / ∂(u,v) =

| ∂x/∂u ∂x/∂v |

| ∂y/∂u ∂y/∂v |

=

| 2/7 1/7 |

| -1/7 3/7 |

So, |J| = (2/7)(3/7) - (1/7)(-1/7) = 8/49.

Using the change of variables, we get:

∬R (x-2y, 3x-y) dA

= ∬R (u, v) |J| du dv

= ∫[1,4] ∫[2u+1,2u+9] (u,v) (8/49) dv du

= (8/49) ∫[1,4] ∫[2u+1,2u+9] (uv) dv du

= (8/49) ∫[1,4] [(1/2)(2u+1+2u+9)(2u+9-2u-1)] du

= (8/49) ∫[1,4] [(2u+5)(8)] du

= (64/49) ∫[1,4] (2u+5) du

= (64/49) [(u^2/2) + 5u] from 1 to 4

= (64/49) [(32/2+20) - (1/2+5)]

= (64/49) (42)

= 5376/49

Therefore, the value of the given integral is 5376/49.

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Lateral Surface Area of triangle prism = (2a + c) x h

We need to determine the area of all the rectangular sides of a triangular prism in order to calculate its lateral surface area. In this instance, there are two isosceles triangles and one rectangle.

The following formula can be used to determine the triangular prism's lateral surface area:

Lateral Surface Area = Perimeter of Base x Height

The sum of the lengths of each side of a triangular prism forms the base's perimeter.

Since the base is an isosceles triangle in this case, the perimeter can be calculated by dividing the length of the third side by twice the length of one of the equal sides.

We should expect that the foundation of the three-sided crystal has sides of length a, b, and c. We can expect to be that an and b are the equivalent sides, and c is the third side.

Perimeter of Base = 2a + c

The height of the triangular prism is the perpendicular distance between the two parallel bases, which is the length of the rectangular face of the prism. Let's assume that the height of the triangular prism is h.

Now, the lateral surface area of the triangular prism can be found by multiplying the perimeter of the base by the height of the prism:

Lateral Surface Area = Perimeter of Base x Height

Lateral Surface Area = (2a + c) x h

Therefore, to calculate the lateral surface area of the triangular prism, we need to know the length of the sides of the base, c and the height of the prism, h. Once we have these values, we can use the formula to find the lateral surface area.

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A) The derivative of  In(6x⁴y⁵) - cos x⁸ = tan² x - X is [(2x³tan x  x  sec² x - 3x¹⁰y x sin x⁸ - (1/2)y)/(2xy⁴)]

B) The derivative of cosh(xy) - sinh² y² = x is (1 - y²sinh(xy))/(xy - 4y³cosh y²)

Problem 1: Find dy given In(6x⁴y⁵) - cos x⁸ = tan² x - X

To find dy, we need to take the derivative of both sides of the equation with respect to x. This means that we will use the chain rule and product rule of differentiation.

Starting with the left-hand side of the equation, we have:

d/dx [In(6x⁴y⁵)] - d/dx [cos x⁸] = d/dx [tan² x - x]

Using the chain rule, we can simplify the first term on the left-hand side as follows:

d/dx [In(6x⁴y⁵)] = (1/(6x⁴y⁵))  x  d/dx [6x⁴y⁵]

= (1/(6x⁴y⁵))  x  [6x⁴  x  d/dx(y⁵) + 5y⁵  x  d/dx(6x⁴)]

= (1/(x⁴y))  x  [4xy⁴  x  dy/dx + 30x³y⁵]

For the second term on the left-hand side, we can simply use the chain rule to get:

d/dx [cos x⁸] = -sin x⁸  x  d/dx [x⁸]

= -8x⁷sin x⁸

For the right-hand side, we can use the power rule and chain rule to get:

d/dx [tan² x - x] = 2tan x  x  sec² x - 1

Now, we can substitute all of these derivatives back into the original equation and solve for dy:

(1/(x⁴y))  x  [4xy⁴  x  dy/dx + 30x³y⁵] - 8x⁷sin x⁸ = 2tan x  x  sec² x - 1

Multiplying both sides by (x⁴y), we get:

4xy⁴  x  dy/dx + 30x³y⁵ - 8x¹¹y x sin x⁸ = (2x⁴y)tan x  x  sec² x - x⁴y

Now, we can solve for dy:

dy/dx = [(2x³tan x  x  sec² x - 3x¹⁰y x sin x⁸ - (1/2)y)/(2xy⁴)]

This is our final answer for dy.

Problem 2: Find dy given cosh(xy) - sinh² y² = x

To find dy, we need to take the derivative of both sides of the equation with respect to x. This means that we will use the chain rule and product rule of differentiation.

Starting with the left-hand side of the equation, we have:

d/dx [cosh(xy)] - d/dx [sinh² y²] = d/dx [x]

Using the chain rule, we can simplify the first term on the left-hand side as follows:

d/dx [cosh(xy)] = y x sinh(xy)  x  d

= ysinh(xy)  x  (ydx/dx + xdy/dx) = y²sinh(xy) + xy x dy/dx

For the second term on the left-hand side, we can use the chain rule and power rule to get:

d/dx [sinh² y²] = 2ycosh y²  x  d/dx [sinh y²] = 4y³cosh y²  x  dy/dx

For the right-hand side, the derivative of x with respect to x is simply 1.

Now, we can substitute all of these derivatives back into the original equation and solve for dy:

y²sinh(xy) + xydy/dx - 4y³ x cosh y²  x  dy/dx = 1

Grouping the terms with dy/dx on one side, we get:

dy/dx  x  (xy - 4y³cosh y²) = 1 - y²sinh(xy)

Dividing both sides by (xy - 4y³ x cosh y²), we get:

dy/dx = (1 - y²sinh(xy))/(xy - 4y³cosh y²)

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The pet food manufacturer is considering adding new kibble mixes to its line of dry dog foods, and they want to test their appeal before introducing them to the market.

They prepared four different kibble mixes, each with a predominant flavor:

Salmon, Turkey, Chicken, and Beef.
The manufacturer collaborated with a local animal shelter to conduct the study.
They randomly divided 64 dogs at the shelter into four different groups, assigning one kibble mix to each group.
At mealtime, each dog was given a serving of their assigned kibble mix.
After the dogs finished eating, the amount of food each dog ate was measured to evaluate the appeal of each kibble mix.
By analyzing the results of this study, the pet food manufacturer can determine which kibble mix is the most appealing and make an informed decision on which new flavors to introduce to their dry dog food line.

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Answer: He has 4 apples left.

Step-by-step explanation:

5-1=4

(hope this helps)

Answer:4

Step-by-step explanation: James had 5 apples which later he gave one to aliya so 5-1 is 4

The roots of the quadratic function, y = -x² - 2x + 8, are -4 and 2. The maximum point is y = 9

From the question, we have a diagram that shows the graph of a quadratic function.

The given quadratic function is

y = -x² - 2x + 8

Since the coefficient of x² is negative as, the graph will open downwards as shown.

To determine the root of a quadratic function from the given graph, we need to find the x-intercepts of the graph. x-intercepts are the points where the graph crosses the x-axis.

From the given graph,

The coordinates of the x-intercepts of the quadratic function are (-4, 0) and (2, 0).

The x-coordinates of these coordinates are -4 and 2.

Thus,

The roots of the quadratic function are -4 and 2

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Answer:

Step-by-step explanation:

So what you want to do is go look at your notes and ten resolve hope this helps!!!

Required values of all angles and sides are 30°, 60°, 90°, 1, 1, √2.

What are the Trigonometric ratios?

[tex]sin(α) = \frac{k}{r} \\ cos(α) = \frac{h}{r} \\ sin(β) = \frac{k}{r} \\ cos(β) = \frac{h}{r} [/tex]

From the given information, we also have:

cos(β) = √3/2

Therefore, we can solve for the remaining values as follows:

[tex]sin(β) = \frac{k}{r} = sin(α) = \sqrt(1 - cos^2(α))[/tex]

√(1 - cos²(α)) = √3/2

1 - cos²(α) = 3/4

cos²(α) = 1/4

cos(α) = ±1/2

Since α is the angle between the hypotenuse and the base, and it is acute, we have:

cos(α) = h/r > 0

Therefore, cos(α) = 1/2

This means that α = 60°.

We can now use the relationships we derived earlier to find the values of k, h, and r:

sin(α) = k/r = √(1 - cos²(α)) = √3/2

k = r√3/2

cos(α) = h/r = 1/2

h = r/2

Using the Pythagorean theorem, we can also find the value of r:

r² = h² + k²

r² = (r/2)² + (r√3/2)²

r² = r²/4 + 3r²/4

r² = r²

r = √4/4 = 1

Therefore, the triangle has side lengths of h = 1/2, k = √3/2, and r = 1, and angles α = 60°, β = arccos(√3/2) = 30°, and 90°.

So, the correct answer is B) 30°, 60°, 90°, 1, 1, √2.

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Answer: the point of inflection is (7, 0).

Step-by-step explanation:

Given function f(x) = (7-x)e^-x

To find the intervals of increase or decrease, we need to find the first derivative of the function and then determine where it is positive or negative:

f'(x) = -e^-x(x-6)

Now, we can use the first derivative test to find the intervals of increase and decrease:

When x < 6, f'(x) is negative, so f(x) is decreasing on the interval (-∞, 6).

When x > 6, f'(x) is positive, so f(x) is increasing on the interval (6, ∞).

Therefore, the intervals of increase and decrease are:

increasing on (6, ∞)

decreasing on (-∞, 6)

To find the intervals of concavity, we need to find the second derivative of the function and then determine where it is positive or negative:

f''(x) = e^-x(x-7)

Now, we can use the second derivative test to find the intervals of concavity:

When x < 7, f''(x) is positive, so f(x) is concave up on the interval (-∞, 7).

When x > 7, f''(x) is negative, so f(x) is concave down on the interval (7, ∞).

Therefore, the intervals of concavity are:

concave up on (-∞, 7)

concave down on (7, ∞)

To find the point of inflection, we need to find where the concavity changes. In this case, the concavity changes at x = 7, so the point of inflection is:

(x, y) = (7, (7-7)e^-7) = (7, 0)

Therefore, the point of inflection is (7, 0).

In interval notation:

Increasing: (6, ∞)

Decreasing: (-∞, 6)

Concave up: (-∞, 7)

Concave down: (7, ∞)

Point of inflection: (7, 0)

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We can say with 90% confidence that the true mean GPA of all accounting students at this university lies between 2.7107 and 3.0493.

We are given:

Sample size n = 21

Sample mean X = 2.88

Sample standard deviation s = 0.16

Confidence level = 90% or α = 0.10 (since α = 1 - confidence level)

Since the sample size is small and population standard deviation is unknown, we will use a t-distribution to construct the confidence interval.

The formula for the confidence interval is given by:

X ± t(α/2, n-1) * s/√n

where t(α/2, n-1) is the t-score with (n-1) degrees of freedom, corresponding to the upper α/2 percentage point of the t-distribution.

Using a t-table with (n-1) = 20 degrees of freedom and α/2 = 0.05, we find the t-score to be 1.725.

Plugging in the values, we get:

2.88 ± 1.725 * 0.16/√21

= (2.7107, 3.0493)

Therefore, we can say with 90% confidence that the true mean GPA of all accounting students at this university lies between 2.7107 and 3.0493.

Note: The confidence interval can also be written as [2.71, 3.05] rounding to two decimal places.

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The RSA algorithm is commonly used to encrypt PGP (Pretty Good Privacy) email messages.

The RSA (Rivest-Shamir-Adleman) algorithm is a widely used asymmetric encryption algorithm that is commonly used for encrypting and decrypting PGP email messages. Asymmetric encryption involves the use of a pair of keys, a public key and a private key. The public key is used for encrypting messages, while the private key is used for decrypting messages. The RSA algorithm uses a complex mathematical process involving prime numbers to generate these keys.

When a PGP email message is encrypted using RSA, the recipient's public key is used to encrypt the message, making it unreadable to anyone who does not possess the corresponding private key. The encrypted message can only be decrypted by the recipient using their private key. This ensures that only the intended recipient can read the contents of the email.

Therefore, the RSA algorithm is commonly used to encrypt PGP email messages, ensuring their confidentiality and security during transmission.

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a)

u = 140 grams

σ = 13 grams

b)

K = 27.15 grams

c)

L1  = 272.85 grams

L2 = 327.15 grams

We have,

(a)

Let X be the weight of strawberries and Y be the weight of blueberries in a fruit cup.

Then we have:

E(X) = 160 grams

SD(X) = 10 grams

E(X+Y) = 300 grams

SD(X+Y) = 15 grams

Since X and Y are independent, we have:

E(X+Y) = E(X) + E(Y) = 160 + u

SD(X+Y) = sqrt(SD(X)^2 + SD(Y)^2) = sqrt(10^2 + σ^2)

Substituting the given values, we get:

160 + u = 300

√(10^2 + o^2) = 15

Solving for u and o, we get:

u = 140 grams

σ = √(15² - 10²) = 13 grams (rounded to the nearest gram)

(b)

Since the weights of fruit cups are normally distributed with mean 300 grams and standard deviation 15 grams, we can find the value of K using the standard normal distribution table.

We want the middle 96.6% of the distribution, which corresponds to a z-score of ±1.81.

Therefore, we have:

K = 1.81 x 15 = 27.15 grams (rounded to the nearest gram)

(c)

We can use the same approach as in part (b) to find the values of L1 and L2. We want the middle 96.6% of the distribution, which corresponds to a z-score of ±1.81.

Therefore, we have:

L1 = 300 - 1.81 x 15 = 272.85 grams (rounded to the nearest gram)

L2 = 300 + 1.81 x 15 = 327.15 grams (rounded to the nearest gram)

Thus,

a)

u = 140 grams

σ = 13 grams

b)

K = 27.15 grams

c)

L1  = 272.85 grams

L2 = 327.15 grams

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Hi!

How to remove the factor from the radical?

The answer is in the picture.

⇒ [tex]\sqrt{75}=\bf\red{\boxed{5\sqrt3}}[/tex]

The cost of the car in 2000 would be $17,520, which is more than four times the original cost in 1970.

To calculate the cost of the car in 2000, we need to first adjust the original cost for inflation. Inflation is the general increase in prices of goods and services over time. So, if the inflation rate is constant at 4%, the cost of the car in 2000 will be much higher than its original cost in 1970.

To calculate the cost of the car in 2000, we can use the formula:

Adjusted cost = Original cost x (1 + Inflation rate)^Number of years

In this case, the original cost of the car in 1970 was $4000, and the inflation rate is constant at 4%. The number of years between 1970 and 2000 is 30.

So, the adjusted cost of the car in 2000 can be calculated as follows:

Adjusted cost = $4000 x (1 + 0.04)^30
Adjusted cost = $4000 x (1.04)^30
Adjusted cost = $4000 x 4.38
Adjusted cost = $17,520

Therefore, the cost of the car in 2000 would be $17,520, which is more than four times the original cost in 1970. This example shows how inflation can have a significant impact on the cost of goods and services over time. It is important to consider inflation when making financial decisions, such as budgeting, saving, and investing.

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In linear equation, The maximum height reached by the rocket, to the nearest tenth of a foot is 503 feet.

What is a linear equation in mathematics?

A linear equation in algebra is one that only contains a constant and a first-order (direct) element, such as y = mx b, where m is the pitch and b is the y-intercept.

                         Sometimes the following is referred to as a "direct equation of two variables," where y and x are the variables. Direct equations are those in which all of the variables are powers of one. In one example with just one variable, layoff b = 0, where a and b are real numbers and x is the variable, is used.

y=-16x²+ 72x + 144

dy/dx = -16(2x)+ 72

Substitute the value of dy/dx as 0, to get the value of x,

0 = -32x + 72

72 = 32x

x = 2.25

Substitute the value of x in the equation to get the maximum height,

y=-16x²+228x+71

y=-16(2.25²)+228(2.25)+71

y= 503 feet

Hence, the maximum height reached by the rocket, to the nearest tenth of a foot is 503 feet.

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Sequence  Explicit Formula Exponential Function  Constant Ratio y-Intercept

A                  -2*3^x-1                f(x) = (-2)3^(x-1)              3                   (0, -2)

B                  45*2^x-1               f(x) = (45)2^(x-1)             2                   (0, 45)

C                 1234*0.1^x-1       f(x) = (1234)0.1^(x-1)          0.1               (0, 1234)

D               -5*(1/2)^x-1             f(x) = -5*(1/2)^(x-1)           1/2              (0, -5)

When a mathematical function can be represented as f(x) = r^x or a^x, where r, or a is the constant and x is the exponent, variable, we call it an Exponential Function. The variable "x" must only appears in the exponent of exponential functions; it does not appear in the base or as a coefficient.

You should know that a function's growth or decline occurs at a constant rate since an exponential function's rate of change is proportionate to the function's value.

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The parameter of interest is the true mean number of calories served at lunch at all schools in the country.

b. Null hypothesis: The true mean number of calories served at lunch at all schools in the country is 880 calories or more. Alternative hypothesis: The true mean number of calories served at lunch at all schools in the country is below 880 calories.

c. A Type I error in this problem would be if the researcher concludes that the mean number of calories served at lunch is below 880 calories when in fact it is 880 calories or more. This means rejecting the null hypothesis when it is actually true.

d. A Type II error in this problem would be if the researcher concludes that the mean number of calories served at lunch is 880 calories or more when in fact it is below 880 calories. This means failing to reject the null hypothesis when it is actually false.

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The given point does not satisfies the equation of the given plane. Therefore, the line does not lie in the plane.

First, let's rearrange the equation of the plane to the standard form Ax + By + Cz = D:

y + (1/8)x + 9y = 10

Simplifying, we get:

(1/8)x + 10y = 10

Multiplying by 8 to eliminate the fraction, we get:

x + 80y = 80

Now let's write the equation of the line in parametric form:

x = t

y = -8t + 9

z = t + 4

Substituting these equations into the equation of the plane, we get:

x + 80y = 80

t + 80(-8t + 9) = 80

Simplifying, we get:

641t = 560

t = 560/641

Substituting this value of t back into the equations of the line, we get:

x = 560/641

y = -8(560/641) + 9

z = 560/641 + 4

x ≈ 0.874

y ≈ 9.76

z ≈ 4.874

So the line intersects the plane at the point (0.874, 9.76, 4.874).

To determine if the line lies in the plane, we need to check if all points on the line satisfy the equation of the plane. Let's substitute the parametric equations of the line into the equation of the plane:

y + (1/8)x + 9y = 10

-8t + 9 + (1/8)t + 9(-8t + 9) = 10

-63t + 81 = 10

-63t = -71

t = 71/63

Substituting this value of t back into the parametric equations of the line, we get:

x = 71/63

y = -8(71/63) + 9

z = 71/63 + 4

x ≈ 1.127

y ≈ 8.111

z ≈ 4.127

As we can see, this point does not satisfy the equation of the plane. Therefore, the line does not lie in the plane.

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For a continuous random variable, X, the value of mean and the variance are equal to the two and eight respectively.

A continuous random variable is defined as a random variable that can posses an infinite number of possible values. Let X be a continuous random variable with a probability density function,[tex]f(x) = \[ \begin{cases} \frac{x}{8}&0< x < 4 \\ 0 & otherwise\end{cases} \][/tex]. We have to determine the mean and the variance for X. We use probability density function, f(x). for determining the mean and variance. The mean of a continuous random variable can be written as [tex]E( X) = \int_{- ∞}^{∞} x f(x) dx [/tex], In this case mean of random variable X is written by [tex]E( X)= \int_{-∞}^{0} x f(x) dx + \int_{0}^{4} x f(x)dx + \int_{4}^{∞} x f(x) dx[/tex]

Substitute the value of function f(x),

[tex] = \int_{0}^{4} x (\frac{x}{8}) dx [/tex]

[tex] = \int_{0}^{4} (\frac{x²}{8}) dx[/tex]

[tex] = [\frac{x^{3} }{8 \times 3}]_{0}^{4}[/tex]

[tex] = [\frac{4³}{8×3} - 0][/tex]

= 2

The variance of a continuous random variable can be defined as the expectation of the squared differences from the mean. The variance of random variable is written as Var( X) = xE(x) ,

= [tex] \int_{-∞}^{∞} x^{2} f(x) dx [/tex]

[tex] = \int_{-∞}^{0} (\frac{x ^{3} }{8}) + \int_{0}^{4} (\frac{x ^{3} }{8}) +\int_{4}^{∞} (\frac{x ^{3} }{8})dx[/tex]

[tex]= [\frac{x^{4} }{8 \times 4}]_{0}^{4} [/tex]

[tex]= [\frac{4⁴}{8×4} - 0][/tex]

= 8

Hence, required variance value is 8.

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Complete question:

Let X be a continuous random variable with a probability density function

[tex]f(x) = \[ \begin{cases} \frac{x}{8}&0< x < 4 \\ 0 & otherwise\end{cases} \][/tex]

Find the mean and the variance for X.

The spherical coordinates are (1, π/3, π/4) with cylindrical coordinates (r,θ,z) So, the correct option is (a) (1, π/3, π/4).

We can use the following relationships between cylindrical and spherical coordinates:

p = √(r² + z²)

θ = θ

φ = arctan(z/r)

Substituting the given values, we get:

p = √(r² + z²) = √((√2/2)²+ (√2/2)²) = 1

θ = π/3

φ = arctan(z/r) = arctan(√2/2 / √2/2) = arctan(1) = π/4

Therefore, the spherical coordinates are (1, π/3, π/4), So, the correct option is (a) (1, π/3, π/4).

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Find the spherical coordinates (p,θ, O ) of the the point with cylindrical coordinates (r,θ,z): (√2/2, π/3,√2/2)

(a) (1, π/3, π/4)

(b)  (1, π/3, √2/2)

(c) (√2/4, √6/4, √2/2)

(d) (√2/4, √6/4, 1)

(e) (√2/4, √6/4, π/4)

(f) None of these

Using inequalities, we can find that the 1st, 2nd and 4th inequality is correct. They are as follows:

1.2 < √5/2 < 1.8

√3/2 < √2 < 1.5

√6 < 2.5 < √7

When utilising the "equal to" symbol in mathematics, equations are not necessarily balanced on both sides. When one thing is superior to or inferior to another, the relationship is commonly referred to as "not equal to". A link between two numbers or other mathematical expressions that leads to an unfair comparison is referred to as an inequality in mathematics. In algebra, inequalities are a particular kind of mathematical expression.

Here in the question:

1st inequality given:

1.2 < √5/2 < 1.8

Now the value of √5/2

= 2.23/2

= 1.11

So, the inequality is correct.

Next, we have:

√3/2 < √2 < 1.5

Now, value of √3/2

= 1.73/2

= 0.86

Value of √2

= 1.41

So, the inequality is correct.

Next, we have:

2.1 < 2.3 < √5

Value of √5

= 2.23

So, this inequality is incorrect.

Next, we have:

√6 < 2.5 < √7

Value of √6

= 2.44

Value of √7

= 2.64

So, the inequality is correct.

Finally, we have:

√3 < 1.8 < √7/2

Now value of √3

= 1.73

Now value of √7/2

= 2.64/2

= 1.32

So, this inequality is incorrect.

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