Find the maximum and minimum values achieved by f(x) =x3 − 9x2 + 15x + 18 on the interval [0,6]

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

To find the area under the curve y = 6x^2 + 4x + 3e^(x/3) from x = 0 to x = 3, we need to integrate the function over the given interval:

∫[0,3] (6x^2 + 4x + 3e^(x/3)) dx

Using the power rule of integration and the exponential rule, we have:

∫[0,3] (6x^2 + 4x + 3e^(x/3)) dx = 2x^3 + 2x^2 + 9e^(x/3) |[0,3]

Plugging in the limits of integration, we have:

(2(3)^3 + 2(3)^2 + 9e^(3/3)) - (2(0)^3 + 2(0)^2 + 9e^(0/3))

= 54 + 9e - 0 - 9

= 45 + 9e

Therefore, the area under the curve from x = 0 to x = 3 is 45 + 9e.

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1. The function f(x) = 5x + 5x⁻¹ has a local maximum at x = √3 with a value of 5√3 + 5/√3 and a local minimum at x = 1/√3 with a value of 5/√3 + 5√3.

2. The concentration of a drug t hours after being injected is given by C(t) = 0.8t / (t² + 6). The maximum concentration occurs at t ≈ 1.63 hours.


1. To find the local maximum and minimum, take the derivative of f(x) and set it to 0. Solve for x to find critical points. Plug these values back into the original function to find the corresponding y-values.

2. For the drug concentration, take the derivative of C(t) and set it to 0. Solve for t to find the time when the concentration is at a maximum. Round your answer to 2 decimal places.

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The probability of C and D being mutually exclusive is not possible because they occur  independently  with a probability of 0.18.

The formula for evaluating the probability of independent events is

[tex]P(A and B) = P(A) * P(B)[/tex]

Then

P(A) and P(B) =  probabilities of events A and B respectively.

For the given case,

we keep two independent events C and D with probabilities

Here,

P(C) = 0.3

P(D) = 0.6

Then, the evaluated  probability of both events occurring together is

[tex]P(C and D) = P(C) * P(D)[/tex]

          [tex]= 0.3 * 0.6[/tex]

          = 0.18

Then, C and D are not mutually exclusive events the reason behind it is  they can occur independently of each other with a probability of 0.18.

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Answer:

Yes that answer is correct

Step-by-step explanation:

Im literally god

a) The probability of X being less than 0.5 is approximately 0.1719.

b) The probability of X being between 0.4 and 0.7 is approximately 0.2531.

c) The expected value of X is 1.25.

d) The cumulative distribution function of X is

First, we need to ensure that f(x) is non-negative for all values of x. Since both 1/8 and 3/8x are non-negative, their sum is also non-negative, and thus f(x) is non-negative for all values of x in the range [0,2].

Second, we need to ensure that the integral of f(x) over the entire range of x equals 1. That is, we need to check that ∫₀² f(x)dx = 1.

∫₀² f(x)dx = ∫₀² (1/8 + 3/8x)dx = (1/8)x + (3/16)x² |0² = (1/8)(2) + (3/16)(2²) - 0 = 1.

Since f(x) satisfies both properties, we can conclude that it is indeed a probability density function.

Next, let's find the probability P(X < 0.5). To do so, we need to integrate f(x) over the range [0,0.5]:

P(X < 0.5) = [tex]\int _{0}^{0.5}[/tex]f(x)dx = [tex]\int _{0}^{0.5}[/tex] (1/8 + 3/8x)dx = (1/8)(0.5) + (3/16)(0.5²) = 0.171875.

Now, let's find the probability P(0.4 < X < 0.7). To do so, we need to integrate f(x) over the range [0.4,0.7]:

P(0.4 < X < 0.7) = [tex]\int _{0.4}^{0.7}[/tex] f(x)dx = [tex]\int _{0.4}^{0.7}[/tex] (1/8 + 3/8x)dx = (1/8)(0.3) + (3/16)(0.7² - 0.4²) = 0.253125.

Next, let's find the expected value (mean) of X. The expected value of a continuous random variable is defined as the integral of x times its PDF over the range of x. That is:

E[X] = ∫₀² xxf(x)dx = ∫₀² x(1/8 + 3/8x)dx = (1/8)(1/2) + (3/8)(1/3)(2³ - 0) = 5/4.

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The median number of siblings for the ten students is: (2 + 3) / 2 = 2.5

To find the median, we first need to arrange the data in order from smallest to largest:

1, 1, 2, 2, 2, 3, 3, 3, 4, 6

The middle value of the data set is 2, since there are five values on either side. Thus, the median number of siblings for the ten students is 2.

Since there are an even number of values, the median is the average of the two middle values, which are 2 and 3. Therefore, the median number of siblings for the ten students is:

(2 + 3) / 2 = 2.5

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The process of rewriting an expression such as -3x²+6x+14 in the form (X+a)² -b is known as completing the square.

An algebraic trick known as "completing the square" is used to change quadratic expressions into a certain form that is simpler to factor or solve for the variable.

These steps are used to square a quadratic expression of the type ax² + bx + c. If the coefficient of x² is not equal to 1, divide both sides of the equation by a.

The equation's constant term (c/a) should be moved to the right side.

To the left side of the equation, add and subtract (b/2a)². The phrase "completing the square" is used to describe it. Consider the left side of the equation to be a perfect square trinomial and factor it.

Find x by taking the square root of either side of the equation.

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The length AC of the missing sides of the given triangle above would be;

AC= 15.6cm

AB = 9cm

To calculate the length of the missing sides of the triangle, the sin rule of used such as;

a/sinA = b/sinB

a = 18

A = 90°

b =AB= ?

B = 30°

That is:

18/sin90 = b/sin 30°

b = 18×0.5/1

= 9

Using Pythagorean formula;

c² = a² +b²

18² = 9²+b²

b² = 324-81

b² = 243

b = √243

= 15.6cm

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False. The general form for a linear equation is given as y = mx + b, where m is the slope and b is the y-intercept.

In a linear equation, the variable y represents the dependent variable and x represents the independent variable. The slope, denoted by m, represents the rate of change of y with respect to x. It determines how steep or flat the line is. The y-intercept, denoted by b, represents the value of y when x is equal to 0, or the point where the line crosses the y-axis.

The correct general form for a linear equation is y = mx + b, not y = a + bx as mentioned in the statement. The slope, denoted by m, multiplies the x variable, and the y-intercept, denoted by b, is a constant that is added or subtracted from the result.  

Therefore, the correct general form of a linear equation is y = mx + b

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a. Based on the data provided, the frequency distribution for question 1 (Gender) is as follows:

Gender
1 - Male: 1
2 - Female: 2

The frequency distribution for question 2 (Ethnicity) is as follows:

Ethnicity
1 - White: 1
2 - Hispanic/Latino: 0
3 - African American/Black: 1
4 - Asian/Pacific Islander: 1

The mean and standard deviation for question 3 (Age) are 20.33 and 1.25 respectively. The mean and standard deviation for question 4 (GPA) are 3.49 and 0.41 respectively.

b. Participants

Method: A survey was conducted to collect data on participants' demographic information, depression levels, and productivity.

Participants: The sample consisted of 3 participants, 2 females, and 1 male, with ages ranging from 19 to 22 years old (M = 20.33, SD = 1.25). The ethnicities represented in the sample were White (n = 1), African American/Black (n = 1), and Asian/Pacific Islander (n = 1). The participants' GPAs ranged from 3.0 to 3.9 (M = 3.49, SD = 0.41).

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The coefficients A and B are determined by the initial conditions of the differential equation. Therefore, the solutions are not based on luck, but on a rigorous mathematical derivation. The given statement is false.

The solution candidates y1(t)=A[tex]e^{(\alpha t)[/tex]cos(βt) and y2(t)=B[tex]e^{(\alpha t)[/tex]sin(βt) for a second-order linear differential equation with constant coefficients and complex roots r1,2=α±βi are not based on pure luck. They are derived using the fact that complex exponential functions can be written as a linear combination of real exponential functions and trigonometric functions through Euler's formula:

e^(α+βi)t = e^αt(cos(βt) + i sin(βt))

Taking the real and imaginary parts of this equation, we get:

e^(αt)cos(βt) = Re(e^(α+βi)t) and e^(αt)sin(βt) = Im(e^(α+βi)t)

So, the solutions y1(t) and y2(t) can be written as linear combinations of exponential functions and trigonometric functions. The coefficients A and B are determined by the initial conditions of the differential equation. Therefore, the solutions are not based on luck, but on a rigorous mathematical derivation.

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Answer:

Step-by-step explanation:

The value of z0 that satisfies P(z ≤ z0) = 0.2348 is 0.73.

Firstly, it's important to note that a standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. This means that all standard normal random variables have the same distribution, regardless of the mean and standard deviation of the original distribution. The standard normal distribution is often used in statistical analysis because it simplifies calculations and allows for easier comparisons between different sets of data.

Now, let's look at the given probability: P(z ≤ z0) = 0.2348. This probability tells us the likelihood of a standard normal random variable z being less than or equal to a certain value z0. To find the value of z0 that satisfies this probability, we can use a standard normal distribution table, which lists the probabilities for different values of z.

Using the table, we can find the closest probability to 0.2348, which is 0.2357. This probability corresponds to a z-value of 0.73

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Complete Question:

Find a value of the standard normal random variable z, call it zo, such that the following probabilities are satisfied.

a. P(z ≤ z0) = 0.2348

The lemniscate is a polar curve given by the equation [tex]r^2 = 11[/tex] sin 2θ.

To find the area inside one loop of the lemniscate, we need to evaluate the definite integral of (1/2)[tex]r^2[/tex]dθ, where [tex]r^2[/tex] is the equation of the curve and we integrate over one full loop, i.e., from 0 to π.

Substituting[tex]r^2[/tex] = 11 sin 2θ, we have:

A = (1/2) ∫[0,π] [tex]r^2[/tex]dθ

= (1/2) ∫[0,π] 11 sin 2θ dθ

Using the trigonometric identity sin 2θ = 2 sin θ cos θ, we can rewrite this integral as:

A = (11/2) ∫[0,π] sin θ cos θ dθ

= (11/4) ∫[0,π] sin 2θ dθ

Integrating sin 2θ with respect to θ from 0 to π, we get:

A = (11/4) [-cos 2θ/2] [0,π]

= (11/4) [-cos π + cos 0]

= (11/2)

Therefore, the area inside one loop of the lemniscate [tex]r^2[/tex]= 11 sin 2θ is (11/2) square units.

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Answer:

[tex]r = \sqrt{ {(6 - 2)}^{2} + {(1 - ( - 3))}^{2} } [/tex]

[tex]r = \sqrt{ {4}^{2} + {4}^{2} } = \sqrt{16 + 16} = \sqrt{32} [/tex]

So the equation of the circle is

[tex] {(x - 2)}^{2} + {(y + 3)}^{2} = 32[/tex]

The solution of given expression having scientific notation is 40,000 by using power formula

Scientific notation is a way of writing numbers that are either very large or very small using powers of 10. It is expressed as a number between 1 and 10 multiplied by a power of 10.

An expression is a mathematical statement that contains numbers, variables, and/or operators. It can be a single number, a combination of numbers and operators, or a combination of numbers, variables, and operators.

To evaluate this expression, we can use the rules of scientific notation to multiply the numbers and then simplify the result.

(4.8 x [tex]10^{8}[/tex]) / (1.2 x [tex]10^{4}[/tex]) x (2.2 x [tex]10^{-6}[/tex])

= (4.8 / 1.2) x [tex]10^{8-4-6+6}[/tex] (multiply the numbers and add the exponents)

= 4 x [tex]10^{4}[/tex]

= 40,000

Therefore, the value of the expression is 40,000.

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The slope of the tangent line at x = 0.4 is approximately 53.96.

To find y', we will use the power rule of differentiation, which states that for any constant n, d/dx(x^n) = nx^(n-1).

So, [tex]y = (x^3 + 3x + 4)^4[/tex]

[tex]y' = 4(x^3 + 3x + 4)^3 * (3x^2 + 3)[/tex]

Now, to find the slope of the tangent line at x = 0.4, we need to evaluate y' at x = 0.4.

[tex]m = y'(0.4) = 4(0.4^3 + 3(0.4) + 4)^3 * (3(0.4)^2 + 3)[/tex]

m = 53.96 (rounded to 1 decimal place)

Therefore, the slope of the tangent line at x = 0.4 is approximately 53.96.

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The antiderivative of   [tex]f(x) = 8x^9 - 3x^6+ 12x^3[/tex] is:  

[tex]F(x) = 4/5 x^{10} - 3/7 x^7+ 3x^4+ C[/tex]

To discover the antiderivative of f(x) = 8x⁹ - 3x⁶ + 12x³, we want to discover a function F(x) such that F'(x) = f(x).

The use of the power rule of integration, we are able to integrate each term of the feature as follows:

[tex]∫(8x^9)dx = (8/10)x^{10}+ C_1 = 4/5 x^{10} + C_1[/tex]

[tex]∫(-3x^6)dx = (-3/7)x^7 + C_2[/tex]

[tex]∫(12x^3)dx = (12/4)x^4+ C_3= 3x^4 + C_3[/tex]

Where the[tex]C_1, C_2, and C_3[/tex] are constants of integration.

Therefore, the antiderivative of [tex]f(x) = 8x^9 - 3x^6 + 12x^3 is:[/tex]

[tex]F(x) = 4/5 x^{10} - 3/7 x^7+ 3x^4+ C[/tex]

Wherein [tex]C = C_1 + C_2 + C_3[/tex] is the steady of integration.

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The solution to the initial value problem is:
y(x) = 2ln(2) + 2ln|x| - y(x)ln|x|.

To solve the initial value problem, we first need to identify the given terms. The problem is given as:

xy'(x) - y(x) = -2, with y(2) = 0

Step 1: Separate variables by dividing both sides by x and y(x), then integrate:

(dy/dx - y/x) = -2/x

(dy/dx) - (y/x) = -2/x

Step 2: Integrate both sides:

∫[1 - (1/x)] dy = ∫(-2/x) dx

Integrating, we get:

y(x) - y(x)ln|x| = -2ln|x| + C

Step 3: Apply the initial condition y(2) = 0:

0 - 0*ln(2) = -2ln(2) + C

C = 2ln(2)

Step 4: Substitute C back into the equation:

y(x) - y(x)ln|x| = -2ln|x| + 2ln(2)

The solution to the initial value problem is:

y(x) = 2ln(2) + 2ln|x| - y(x)ln|x|.

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To answer this question, we need to use the normal distribution formula and the z-score table.

Therefore, the 4% of items with the shortest lifespan will last less than 1.5 years.

First, we need to find the z-score associated with the 4th percentile (since we are looking for the 4% of items with the shortest lifespan).

Using the z-score table, we find that the z-score associated with the 4th percentile is -1.75.

Next, we use the formula:

z = (x - μ) / σ

where z is the z-score, x is the value we are trying to find (the lifespan we want to know), μ is the mean lifespan (3.1 years), and σ is the standard deviation (0.9 years).

Plugging in the values we know:

-1.75 = (x - 3.1) / 0.9

Solving for x:

x - 3.1 = -1.575

x = 1.525

Therefore, the 4% of items with the shortest lifespan will last less than 1.5 years (to one decimal place).
To determine the lifespan for the bottom 4% of items, we will use the mean, standard deviation, and z-score. The mean lifespan is 3.1 years, and the standard deviation is 0.9 years. Using a z-score table, we find that the z-score for the 4% percentile is approximately -1.75.

Now, we can use the formula:
Lifespan = Mean + (z-score × Standard Deviation)
Lifespan = 3.1 + (-1.75 × 0.9)

Lifespan ≈ 1.5 years

Therefore, the 4% of items with the shortest lifespan will last less than 1.5 years.

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[tex]\textit{area of a triangle}\\\\ A=\cfrac{1}{2}bh ~~ \begin{cases} b=base\\ h=height\\[-0.5em] \hrulefill\\ h=\frac{1}{3}\\[1em] A=\frac{5}{36} \end{cases}\implies \cfrac{5}{36}=\cfrac{1}{2}\cdot b\cdot \cfrac{1}{3}\implies \cfrac{5}{36}=\cfrac{b}{6} \\\\\\ 30=36b\implies \cfrac{30}{36}=b\implies \cfrac{5}{6}=b[/tex]

The curve for the given point is illustrated through the following graph.

Let's start by considering the point (3,0). This means that the curve must pass through the point (3,0). We don't know the shape of the curve yet, but we know that it must go through this point.

We are told that the derivative of the function is negative for x > 3. This means that the function is decreasing in this region. To sketch a curve that satisfies this condition, we can draw a curve that starts at (3,0) and then goes downwards towards negative infinity. We can choose any shape for the curve as long as it satisfies this condition.

We now have two parts of the curve, one that goes downwards from (3,0) and one that goes upwards from (0,0). We need to connect these two parts to get a complete curve. To do this, we can draw a curve that passes through (1,1) and (2,-1), for example. This curve will connect the two parts of the curve we already have and satisfy all the given conditions.

In conclusion, to sketch a curve with the given criteria, we start at (3,0) and draw a curve that goes downwards for x > 3 and upwards for x < 0. We then connect these two parts with a curve that passes through (1,1) and (2,-1). The final curve satisfies all the given conditions.

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The distribution of X, which represents the number of words in a randomly selected romance novel, can be described as a normal distribution with a mean (μ) of 64,143 and a standard deviation (σ) of 17,337. In notation form, it is written as: X ~ N(64,143, 17,337).

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To find the maximum value of the given function, we need to take the derivative of the function and set it equal to zero.

y = [tex]x^{3}[/tex] + 3[tex]x^{2}[/tex]- 45x + 24
y' = 3[tex]x^{2}[/tex] + 6x - 45

Setting y' equal to zero:
3[tex]x^{2}[/tex]+ 6x - 45 = 0

Using the quadratic formula, we get:
x = (-6 ± [tex]\sqrt{( 6^{2} - 4(3)(-45)}[/tex] / (2(3))
x = (-6 ± 18) / 6

x = -3, 5

To determine which value of x gives the maximum value of the function, we need to evaluate the second derivative of the function at each critical point.

y'' = 6x + 6

When x = -3:
y'' = 6(-3) + 6 = -12

When x = 5:
y'' = 6(5) + 6 = 36

Since the second derivative at x = 5 is positive, we know that x = 5 gives the maximum value of the function. Therefore, the maximum value of the function is:

y(5) = [tex]5^{3}[/tex] + 3([tex]5^{2}[/tex]) - 45(5) + 24
y(5) = 124

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The maximum value of y is 100, and it occurs when x = -4.

The maximum value for the given function y = x^3 + 3x^2 - 45x + 24. To find the maximum, we'll need to find the critical points of the function by taking the derivative and setting it equal to zero.

1. Find the derivative of the function with respect to x: y' = 3x^2 + 6x - 45
2. Set the derivative equal to zero and solve for x: 3x^2 + 6x - 45 = 0
3. Factor the quadratic equation: x = (-6 ± sqrt(6^2 - 4(3)(-45))) / (2(3))
4. Further factor the quadratic:  (-6 ± 18) / 6
5. Solve for x: x = -4 or x = 3

Now, we need to determine if these points are maxima or minima by using the second derivative test:

6. Find the second derivative of the function: y'' = 6x + 6
7. Evaluate the second derivative at x = -4 and x = 3:
  y''(-5) = 6(-4) + 6 = -18 (negative value indicates a maximum)
  y''(3) = 6(3) + 6 = 24 (positive value indicates a minimum)

Therefore, we have a maximum at the value of x = -4.

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a) f(x) attains an absolute maximum on the interval [0,1] at x=1.

b) It is decreasing on no interval within [0,1].

c) f(1) ≥ 1, and the smallest p.

a) Yes, by the Extreme Value Theorem, since f(x) is continuous on the

closed interval [0,1], it attains a maximum and minimum on this interval.

Moreover, since the derivative is non-negative, f(x) is increasing on [0,1],

and thus its maximum value is attained at x=1.

Therefore, f(x) attains an absolute maximum on the interval [0,1] at x=1.

b) Since the derivative is non-negative, f(x) is increasing on [0,1]. Therefore, it is decreasing on no interval within [0,1].

c) By the Mean Value Theorem, there exists a point c in the open interval

(0,1) such that:

f'(c) = (f(1) - f(0))/(1-0) = f(1) - 1

Since f'(x) is always non-negative, we have:

f(1) - 1 = f'(c) ≥ 0

Therefore, f(1) ≥ 1, and the smallest p

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Using a linear approximation, the value of f(2.9) is approximately 1.6.

To find an approximation to f(2.9) using a linear approximation, we can use the equation for a tangent line:

L(x) = f(a) + f'(a)(x - a)

Here, f(3) = 2, f'(3) = 4, and a = 3. We want to approximate f(2.9), so x = 2.9. Plugging in these values, we get:

L(2.9) = 2 + 4(2.9 - 3)

L(2.9) = 2 + 4(-0.1)

L(2.9) = 2 - 0.4

L(2.9) = 1.6

So, using a linear approximation, the value of f(2.9) is approximately 1.6.

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The general term of the sequence is given by an = (-1)ⁿ⁺¹*2ⁿ/3ⁿ⁻¹, where n is the term number starting from 1.

This formula is obtained by observing that each term is obtained by multiplying the previous term by -2/3 and changing its sign. This pattern can be represented mathematically using exponents, which results in the given formula.

To find the general formula for a sequence, we need to identify the pattern in the given terms. Here, we observe that each term is obtained by multiplying the previous term by -2/3 and changing its sign. This means that the sequence alternates between positive and negative values, and the magnitude of each term is increasing as n increases.

To represent this pattern mathematically, we can use the concept of exponents. Specifically, we can write the numerator of each term as 2ⁿ⁻¹, since the magnitude of each term is increasing by a factor of 2.

Similarly, we can write the denominator of each term as 3ⁿ⁻¹, since the magnitude of each term is decreasing by a factor of 3. Finally, we need to account for the alternating signs of the terms, which is done using the factor.

Putting all these pieces together, we get the formula an = (-1)ⁿ⁺¹*2ⁿ/3ⁿ⁻¹. This formula gives us the nth term of the sequence, assuming that the pattern of the first few terms continues. We can use this formula to find any term of the sequence, without having to compute all the previous terms.

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We can expect that around 160 seniors at Allie's high school eat ice cream at least once per week. This can be answered by the concept of proportions.

To find the estimated number of seniors who eat ice cream at least once per week, we can use proportions.

First, we know that Allie surveyed 720 seniors and 144 of them said they eat ice cream at least once per week. So the proportion of seniors who eat ice cream at least once per week in Allie's sample is:

144/720 = 0.2

This means that 20% of the seniors in Allie's sample eat ice cream at least once per week.

To estimate the number of seniors who eat ice cream at least once per week in the entire school, we can use this proportion and apply it to the total number of seniors:

0.2 x 800 = 160

So we can expect that around 160 seniors at Allie's high school eat ice cream at least once per week.

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Answer:

C. Ten thousands

Step-by-step explanation:

The Statement asked  based on the Variable is , True.

A variable is a symbol or letter that represents a quantity or value that can vary or change in a given context or problem. Variables can be used to express relationships between quantities or to describe patterns or trends in data. They can be either dependent or independent, depending on the context of the problem. An independent variable is a variable that is changed or controlled by the experimenter or observer, while a dependent variable is a variable that is affected or influenced by the independent variable.

True.

A continuous random variable is one that can take on any value within a certain range or interval, and its value is determined by measurement. Examples of continuous random variables include height, weight, temperature, and time, where the values can be any real number within a certain range.

In contrast, a discrete random variable can only take on certain values, typically integers, and its value is determined by counting. Examples of discrete random variables include the number of heads in a series of coin tosses, the number of cars sold in a day, and the number of defects in a batch of products.

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