Jace ran the 100-yard dash in 13.82 seconds. Amy ran the 100-yard dash in 12.98 seconds. How much faster did Amy run the race than Jace?

Answer:

Missing leg = [tex]2\sqrt{14}[/tex]

Step-by-step explanation:

Pythagorean theorem

[tex]a^{2} +b^{2}=c^{2}[/tex]

[tex]12+b^{2} = 8^{2}[/tex]

[tex]12+b^{2} =64[/tex]

[tex]b^{2}=56\\[/tex]

[tex]b=\sqrt{56}[/tex]

[tex]b=2\sqrt{14}[/tex]

Answer:

  13/32

Step-by-step explanation:

You want the area of the shaded portion of the unit square shown.

Points B, C, E are shown as equidistant from point F, so will lie on a circle centered at F. The center of that circle is at the point of coincidence of the perpendicular bisectors of BE, BC, and CE.

Without loss of generality, we can let line EF lie on the x-axis such that E is at the origin. Chord EB of the circle has a rise of 1/2 for a run of 1, so a slope of 1/2. Its midpoint is (1, 1/2)/2 = (1/2, 1/4). The perpendicular line through this point will have slope -2, so its equation can be written ...

  y -1/4 = -2(x -1/2)

  y = -2x +5/4

Then the x-intercept (point F) will have coordinates (0, 5/8):

  0 = -2x +5/4 . . . . . y=0 on the x-axis

  2x = 5/4

  x = 5/8

Trapezoid EFCD will have upper base 5/8, lower base 1, and height 1/2. Its area is ...

  A = 1/2(b1 +b2)h

  A = (1/2)(5/8 +1)(1/2) = (1/4)(13/8) = 13/32

The shaded area is 13/32.

__

Additional comment

The point-slope equation of a line through (h, k) with slope m is ...

  y -k = m(x -h)

If no two ladies sit together, there are 13,228,800,000 distinct ways to arrange the 11 males and 8 women in a row.

If no 2 women can sit together, then we can think of the men and women as distinct blocks that must alternate. Since there are 11 men and 8 women, there must be 12 positions available for these blocks: M_W_M_W_M_W_M_W_M_W_M_W.

The first block can be filled In 11! Ways (11 men to choose from), the second block can be filled in 8! Ways (8 women to choose from), the third block can be filled in 10! Ways, the fourth block can be filled in 7! Ways, and so on. So the total number of ways to seat the group is:

11! * 8! * 10! * 7! * 9! * 6! * 8! * 5! * 7! * 4! * 6! * 3! * 5! * 2! * 4! * 1! * 3! * 2! * 1!

Simplifying this expression, we get:

(11! * 8! * 10! * 7! * 9! * 6! * 8! * 5! * 7! * 4! * 6! * 3! * 5! * 2! * 4! * 1! * 3! * 2! * 1!) =

(11! * 10! * 9! * 8! * 7! * 6! * 5! * 4! * 3! * 2! * 1!) * (8! * 7! * 6! * 5! * 4! * 3! * 2! * 1!) =

11! * 10! * 9! * 8! * 7! * 6! * 5! * 4! * 3! * 2! * 1! * 8! * 7! * 6! * 5! * 4! * 3! * 2! * 1! =

11! * 10! * 9! * 8! * 7! * 6! * 5! * 4! * 3! * 2! * 8! * 7! * 6! * 5! * 4! * 3! * 2!

= 11! * 10! * 9! * 8! * 7! * 6! * 5! * 4! * 3! * 2! * 8!^2 * 7!

We can evaluate this expression with a calculator to obtain:

(111098765432)(887654327) = 13,228,800,000

Therefore, there are 13,228,800,000 different ways to seat the 11 men and 8 women in a row if no 2 women are to sit together.

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The trigonometric expression cosAcotA = 8x/(4x² - 1)

A trigonometric expression is an expression that contains trigonometric ratios.

Given that for any acute angle A if sin A = 2x-1/2x+1, we desire to find the value of cos A cot A?

So, we proceed as follows

cosAcotA = cosA × cosA/sinA  (since cotA = cosA/sinA)

= cos²A/sinA

Now using the trigonometric identity

sin²A + cos²A = 1

⇒ cos²A = 1 - sin²A

So, substituting this into the equation, we have that

cosAcotA = cos²A/sinA

=  (1 - sin²A)/sinA

= 1/sinA - sin²A/sinA

= 1/sinA - sinA

Substituting the value of sinA into the equation, we have

= 1/(2x - 1)/(2x + 1) - (2x - 1)/(2x + 1)

= (2x + 1)/(2x - 1) - (2x - 1)/(2x + 1)

Taking the L.C.M, (2x - 1)(2x + 1), we have

= [(2x + 1)² - (2x - 1)²]/[(2x - 1)(2x+ 1)]

=  [(2x + 1)² - (2x - 1)²]/[(2x)² - 1²)]

=  [(2x + 1 + 2x - 1)(2x + 1 - (2x - 1)]/(2x)² - 1²)

=  [(2x + 1 + 2x - 1)(2x + 1 - 2x + 1)]/4x² - 1)

=  [(4x)(2)]/4x² - 1)

= 8x/(4x² - 1)

So, cosAcotA = 8x/(4x² - 1)

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The number of participants that will not get two cups of water if Mr. Mohan is the only person that donated water, obtained using basic arithmetic operations are 40 participants

Basic arithmetic operations includes addition, subtraction, division and multiplication.

Mr. Mohan donated 8 five-litre bottles of water, which is equivalent to 40 litres of water;

8 × 5 litres = 40 litres

Each cup contains 125 ml of water which means that each liter can fill 8 cups. So the total number of cups that can be filled from the 8 five-litre bottles of water containing 40 litres of water is; 40 × 8 = 320 cups

Since each participant takes one cup at each stop and there are two stops in total, each participant will take a total of 2 cups during the walkation. This means that only 160 participants will get two cups of water if Mr Mohan is the only person who donated water.

So, 200 - 160 = '40 participants' will not get two cups of water if Mr. Mohan is the only person who donated water.

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The probability that a randomly selected unit will contain at least one surface-finish defect is 0.0952.

The number of surface finish flaws in this issue will follow a Poisson distribution, and since the mean is given, it is necessary to obtain the probability. More solutions will be found by using the Poisson distribution's normal approximation.

The following formula can be used to determine the likelihood that a randomly chosen unit will have at least one surface finish flaw:

The distribution's average and standard deviation are λ = 0.1

P(X ≥ 1) = 1 − P(X < 1)

P(X ≥ 1) = 1 − P(X = 0)

P(X ≥ 1) = 1 − [tex]\frac{e^{-0.1}(0.1)^0}{0!}[/tex]

After simplification

P(X ≥ 1) = 0.0952

​

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When the given system of equations does not have a solution, it implies that there is no common point for both the straight lines to intersect. It means that both the straight lines are parallel to each other.

Therefore, the slopes of both lines are equal but their y-intercepts are different. Thus, we can equate the slopes of both lines in the given system of equations as they are equal. Then, we can form an equation to calculate the value of p.Let the given system of equations be:y =[tex]px + 4......(1)y = px + 6......[/tex](2)Equating the slopes of both lines, we have:p =[tex]p => 2p = 10 => p = 5[/tex]Thus, the value of p is 5. We can verify this by substituting the value of p in both the given equations. Therefore, the given system of equations can be written as:y = 5x + 4......(1)y = 5x + 6......([tex]5x + 4......(1)y = 5x + 6......([/tex]2)On comparing equations (1) and (2), we observe that their y-intercepts are different. The equation (1) has y-intercept 4 and equation (2) has y-intercept 6.

Hence, there is no common point for both the lines to intersect. Therefore, the system of equations has no solution.

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The maximum acceleration attained on the interval 0 < t <3 is 27 m/s²

To find the maximum acceleration of the particle, we need to find its acceleration function by taking the derivative of its velocity function with respect to time

a(t) = v'(t) = 3t^2 - 6t + 12

To find the maximum acceleration attained on the interval 0 < t <3, we need to find the critical points of the acceleration function within the interval and evaluate the function at these points and at the endpoints of the interval.

The critical points of the acceleration function occur where its derivative equals zero

a'(t) = 6t - 6 = 0

t = 1

So the only critical point of the acceleration function within the interval 0 < t <3 is t=1. We also need to evaluate the acceleration function at the endpoints of the interval

a(0) = 12

a(3) = 27

Therefore, the maximum acceleration attained on the interval 0 < t <3 is either at t=0, t=1, or t=3. To determine which one is the maximum, we can evaluate the acceleration function at each of these points and compare the values

a(0) = 12

a(1) = 9

a(3) = 27

Thus, the maximum acceleration attained on the interval 0 < t <3 is 27m/s², which occurs at t=3.

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Answer:

Step-by-step explanation:

fsffd

The five-card hand is dealt at random from a standard 52-card deck. the probability of having exactly 2 kings and the ace of spades given that you have at least one ace is 2.9%.

        To find the probability of having exactly 2 kings and the ace of spades given that you have at least one ace

        When a five-card hand is dealt at random from a standard 52-card deck is as follows:

        There are 52 cards in a standard deck. Therefore, there are 4 Kings and 1 Ace of Spades. The number of ways in which we can select 2 Kings from 4 Kings is:

                             ₄C₂ = [tex](4 * 3)/(2 * 1)[/tex]

                                    = 6

           The number of ways in which we can select the Ace of Spades from 1 Ace is:

                              ₁C₁ = 1

            The remaining 2 cards are non-Kings and non-Ace of Spades cards. So, there are 44 such cards. The number of ways in which we can select 2 cards from 44 cards is:

                           ₄₄C₂ = [tex](44 * 43)/(2 * 1)[/tex]

                                   = 946

      Therefore, the total number of ways in which we can select 2 Kings and 1 Ace of Spades and 2 non-Kings and non-Ace of Spades cards is:

                        [tex]6 * 1 * 946[/tex] = 5676

      There are 4 Aces in a standard deck. The number of ways in which we can select at least one Ace from 4 Aces is:

                 ₄C₁ + ₄C₂ + ₄C₃ + ₄C₄ = 4 + 6 + 4 + 1

                                                    = 15

       Therefore, the probability of having exactly 2 kings and the ace of spades given that you have at least one ace when a five-card hand is dealt at random from a standard 52-card deck is:

P(exactly 2 Kings and the Ace of Spades | at least one Ace) = (Number of ways in which we can select 2 Kings and 1 Ace of Spades and 2 non-Kings and non-Ace of Spades cards) / (Number of ways in which we can select at least one Ace)

                         = 5676/15

                         = 378.4 or 0.029 or 2.9%.

   Therefore, the required probability is 0.029 or 2.9%.

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The type of study which the student is conducting is a survey

A survey is a set of questions used in human subject research with the goal of gathering specific information from a certain population. Surveys can be carried out via the phone, by mail, online, at street corners, and even in shopping centers.

This is because the student stands in the main dining hall during dinner 3 times and asks every 10th student how much he/she exercises in a week, in minutes and this is to get reasonable and valuable statistical feedback from the students

The correct answer is A. Survey

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As per the unitary method, Nikhil has 21 pencils and Cortez has 63 pencils.

Let's say Nikhil has x number of pencils. Then, according to the problem, Cortez has three times as many pencils as Nikhil. Therefore, Cortez has 3x number of pencils.

Together, they have a total of 84 pencils. So, we can write an equation based on the number of pencils owned by Nikhil and Cortez as follows:

x + 3x = 84

Simplifying the equation, we get:

4x = 84

Dividing both sides by 4, we get:

x = 21

So, Nikhil has 21 pencils. Using the fact that Cortez has three times as many pencils as Nikhil, we can find out how many pencils Cortez has:

Cortez has 3x = 3(21) = 63 pencils.

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Complete Question:

Lance has three times as many pencils as Nick, and they have 84 pencils together. How many pencils does each of them have?

Answer: Option A: x=4

The system of equations, helps to determine the quantity of each kind of coffee for mixture. The 9 pounds of Guatemala antigua coffee blend and one pound of gazebo coffee blend should be mix.

We have a coffee distributor needs to mix a(n) Guatemala antigua coffee blend. Let us assume, x = pounds of the Guatemala antigua coffee blend

y = pounds of the gazebo coffee blend

The new mixture is 10 pounds, so x + y = 10

And normally sells value of Guatemala antigua coffee blend per pound = $8.50

so, sells for x pounds of the Guatemala antigua coffee blend = $8.50x

Normally sells value of gazebo coffee blend per pound = $13.10y

Total sells = $12.64 × 10

8.50x + 13.10y = 12.64 × 10

We have the following system of equations:

x + y = 10 --(1)

8.50x + 13.10y = 126.4 --(2)

We have to solve the system of equations for determining value x and y. Using the Elimination method, from equation (1), x = 10 - y, put in equation (2), 8.50x + 13.10y = 126.4

=> 8.50 ( 10 - y ) + 13.10y = 126.4

=> 85 - 8.5y + 13.10y = 126.4

=> 4.6 y = 41.4

=> y = 41.4/4.6 = 9

from equation (1), x = 10 - 9 = 1

Hence, required value is 9 pounds and 1 pounds.

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The expression (1 + 1 / 18)¹⁸ represents the number closest to irrational number e. (Correct choice: A)

This problem ask us to determine what expression is closest to irrational number e. A number is irrational when it cannot be represented by numbers of the form m / n, where m and n are integers and n is non-zero. The number e can be defined by the following limit:

[tex]e = \lim_{x \to \infty} \left(1 + \frac{1}x}\right)^{x}[/tex], where x is a natural number.

According to this definition, the greater the value of x is, the closer the result to irrational number e is. Therefore, the following expression is closest to the given irrational number:

x = 18

e = (1 + 1 / 18)¹⁸

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If the length of a rectangle is nine inches more than its width and its area is 442 square inches, the width of the rectangle is 17 inches and its length is 26 inches.

Let's denote the width of the rectangle as "w" and its length as "l". From the problem statement, we know that:

l = w + 9 (since the length is nine inches more than the width)

The area of the rectangle is given by the formula:

A = l * w

Substituting in the expression for "l" in terms of "w", we get:

A = (w+9) * w

Simplifying, we get a quadratic equation:

w^2 + 9w - 442 = 0

We can solve this equation by factoring or using the quadratic formula. Factoring gives:

(w+26)(w-17) = 0

Therefore, the possible values for "w" are -26 and 17. However, since the width of the rectangle cannot be negative, we can eliminate -26 as a possible value. Thus, the width of the rectangle is 17 inches.

Using the expression for "l" in terms of "w", we can find the length:

l = w + 9 = 17 + 9 = 26 inches

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the given composite shape has a 23 square inch surface area.

The area that any composite shape covers is known as the area of composite shapes. The composite shape is a shape created by joining a small number of polygons to create the desired shape. These shapes or figures can be constructed from a variety of shapes, including triangles, squares, quadrilaterals, etc. To calculate the area of a composite shape, divide it into basic shapes like a square, triangle, rectangle, or hexagon.
A composite shape is essentially a combination of basic shapes. A "composite" or "complex" shape is another name for it.
The dimensions of rectangle ABCD are 2 inches long and 7 inches wide.
The DEFG square's side measures 3 inches.

Using the equation for the composite shape's area,

Area of composite shape = Area of square + Area of rectangle; Area of composite shape = Length, Breadth, and Side 2; Area of composite shape = BC, AB, and DE2; Area of composite shape = 2 x 7 + 32; Area of composite shape = 14 + 9 = 23 square inches.

As a result, the given composite shape has a 23 square inch surface area.
The dimensions of rectangle ABCD are 2 inches long and 7 inches wide.
The DEFG square's side measures 3 inches.

Using the equation for the composite shape's area,

Area of composite shape = Area of square + Area of rectangle; Area of composite shape = Length, Breadth, and Side 2; Area of composite shape = BC, AB, and DE2; Area of composite shape = 2 x 7 + 32; Area of composite shape = 14 + 9 = 23 square inches.

As a result, the given composite shape has a 23 square inch surface area.
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Complete question:- What is the area of the composite shape ?

Move all terms not containing the variable from the center section of the inequality.

Answer:

27..

Step-by-step explanation:

w - 10 ≤ 16

w - 10 + 10 ≤ 16 + 10

w ≤ 26

all apply except 27.

The  least common denominator of 7/9 and 2/3 is 9

To find the least common denominator of 7/9 and 2/3, we need to find the smallest multiple that both denominators share.

The multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, ...

The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...

We can see that the smallest multiple that both denominators share is 9.

To convert 7/9 and 2/3 to have a denominator of 9, we need to multiply the numerator and denominator of 7/9 by 1 (which is equivalent to 9/9), and the numerator and denominator of 2/3 by 3:

7/9 * 1 = 7/9

2/3 * 3/3 = 6/9

Now both fractions have a denominator of 9, so we can compare them easily:

7/9 and 6/9

Therefore, the least common denominator of 7/9 and 2/3 is 9.

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The answer to this question is 0.

To add the additive inverse to the product of - 1 and -4, we must first find the product of -1 and -4. The product of -1 and -4 is 4. Now that we have found the product of -1 and -4, we can add the additive inverse to it. The additive inverse of 4 is -4, so we add -4 to 4. 4+ (-4) is equal to 0. Therefore, the answer to this question is 0.

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The probability that at any given time instant, 4 or more users are transmitting simultaneously is 0.3297

The binomial distribution is the discrete probability distribution used in probability theory and statistics that only allows for Success or Failure as the potential outcomes of an experiment. For instance, if we flip a coin, there are only two conceivable results: heads or tails, and if we take a test, there are only two possible outcomes: pass or fail. A binomial probability distribution is another name for this distribution.

(a) 21 users

No. of users supported = 2.4 Mbps/450 kbps

= (2.4 x 1000000) / ( 450x 1000)

=20

(b)  Probability = 10% of a time

=1/10

=0.1

(c)  By using binomial distribution,

probability for N users = [tex]120 CN* (0.1) ^ N * (9/10) ^ 120 - N[/tex]

(d)  Total number of chances = 17

probability for more than 4 users = 0.6703

So, 4 and more user's probability = 0.3297

=0.3297

The probability of 4 or more users are transmitting simultaneously is 0.3297.

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The perimeter of  isosceles triangle PRQ is 24.65 units.

The lengths of its three sides and add them up. We can use the distance formula to find the length of each side.

Distance formula: The distance d between two points (x1, y1) and (x2, y2) is given by:

d = [tex]sqrt((x2 - x1)^2 + (y2 - y1)^2)[/tex]

Using this formula, we can find the lengths of the sides PR, RQ, and PQ.

PR = [tex]sqrt((-1 - (-7))^2 + (4 - (-1))^2) = sqrt(36 + 25) = sqrt(61)[/tex]

RQ =[tex]sqrt((3 - (-1))^2 + (-1 - 4)^2) = sqrt(16 + 25) = sqrt(41)[/tex]

PQ = [tex]sqrt((3 - (-7))^2 + (-1 - (-1))^2) = sqrt(100) = 10[/tex]

Therefore, the perimeter of triangle PRQ is:

PR + RQ + PQ = [tex]sqrt(61) + sqrt(41) + 10 ≈ 24.65[/tex] (rounded to two decimal places)

So, the perimeter of triangle PRQ is approximately 24.65 units.

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The forecaster (or the software) must set three parameters using Winter's exponential smoothing. (Option 4)

Winter's exponential smoothing is a method for forecasting in which the forecast for the next period is weighted more heavily based on the data that was recorded in previous periods.

The formula for Winter's exponential smoothing includes three parameters: alpha, beta, and gamma. Parameters in Winter's exponential smoothing:

alpha: The degree of smoothing that is applied to the level of the series. The alpha value will vary between 0 and 1.

beta: The degree of smoothing applied to the trend of the series. The beta value will vary between 0 and 1.

gamma: The degree of smoothing applied to the seasonal component of the series. The gamma value will vary between 0 and 1.

Hence, the forecaster (or the software) must set three parameters using Winter's exponential smoothing.

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The bread recipe calls for 2 more cups of flour per cup of sugar than the cookie recipe.

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

Explanation:

I'll use this shorthand

Something like "2 cups of sugar" will shorten to "2s".

The cookies need 3s for every 6f.

Therefore we get this ratio

3s : 6f

And we can divide both sides by 3 to end up with

s : 2f

Meaning "each cup of sugar needs 2 cups of flour".

--------------------

Meanwhile, the bread recipe calls for 2s for every 8f.

2s : 8f

2s/2 : 8f/2 .... divide both sides by 2

s : 4f

When making bread, each cup of sugar needs 4 cups of flour. In other words, the amount of flour is quadruple that of the sugar.

--------------------

Recap

These ratios are reduced so that we have "s" on the left side without any numbers attached to it. In other words, we have 1s.

We can see that the bread needs 2 more cups of flour compared to the cookies (since 4f - 2f = 2f). It's important that we reduce those ratios so that we're talking about the same amount of sugar for each food item.

This is why  choice B   is the final answer.

--------------------

An example:

Each food item will use 1 cup of sugar.

The cookies need twice the amount of flour, so the cookies require 2 cups of flour (because of the ratio s:2f mentioned earlier).

The bread needs quadruple the amount of flour compared to sugar. Meaning the bread needs 4 cups of flour (because of the ratio s:4f).

Then subtract to get

(4 cups flour for bread) - (2 cups flour for cookies) = 2 cups flour

The probability of getting a tail on the coin and at least a 5 on the die is 0.1667 rounded to four decimal places.

The probability is frequently stated as a ratio between the total number of outcomes in the sample space and the number of positive outcomes. This is how it is written: Number of Favorable Outcomes P(E) = Probability of an Event (Sample space).

If a coin is fair and impartial, the likelihood of getting a tail is equal to half.

On a six-sided dice, the likelihood of getting at least a 5 is 2/6, or 1/3. Either rolling a 5 or rolling a 6 will result in a 5 or a 6 on the die. The likelihood of rolling at least a 5 is 2/6 or 1/3 because there are a total of six possible results.

You can multiply the probabilities of the two occurrences to determine the likelihood that they will occur simultaneously (assumed to be independent events). As a result, the likelihood of getting at least a 5 on the die and a tail on the coin is:

1/2 * 1/3 = 1/6

Thus, there is a 1 in 6 possibility of rolling at least a 5 on the die and getting a tail on the coin in a single trial. Another way to say this is as a percentage:

1/6 * 100% = 16.67%

Thus, rounded to four decimal places, the likelihood of receiving a tail on the coin and at least a 5 on the die is 0.1667.

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As per the given probability, the values of

(a) E (X) is 17.4 ft³, E (X²) is 323.2 ft⁶, and V (X) is 2.16 ft⁶.

To calculate the expected value of X (E(X)), we multiply each possible value of X by its probability and add up the results. Using the probability mass function given, we have:

E(X) = (16)(0.3) + (18)(0.5) + (20)(0.2) = 17.4

This means that on average, the rated capacity of a freezer sold at this store is 17.4 ft³.

To calculate the expected value of X squared (E(X²)), we follow a similar process, but instead of multiplying each possible value of X by its probability, we square each possible value of X and multiply it by its probability before adding up the results. Using the probability mass function given, we have:

E(X²) = (16²)(0.3) + (18²)(0.5) + (20²)(0.2) = 323.2

This means that on average, the square of the rated capacity of a freezer sold at this store is 323.2 ft⁶.

To calculate the variance of X (V(X)), we then add up the results to obtain the variance. Using the probability mass function given, we have:

V(X) = (16 - 17.4)²(0.3) + (18 - 17.4)²(0.5) + (20 - 17.4)²(0.2) = 2.16

This means that the variance of the rated capacity of a freezer sold at this store is 2.16 ft⁶. The square root of the variance is known as the standard deviation, which gives us a measure of the spread of the distribution of X.

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Complete Question:

A certain brand of upright freezer is available in three different rated capacities: 16 ft³, 18 ft³ , and 20 ft³ . Let X = the rated capacity of a freezer of this brand sold at a certain store. Suppose that X has the following probability mass function:

x 16 18 20

p(x) 0.3 0.5 0.2

(a) Compute E (X) , E (X²) , and V (X) .

The area of the square is changing at 12√2 + 2√2t square inches/minute when the circle is increasing at 1 in/min.

Let us find the relationship between r and s using the Pythagorean theorem. Since the square is inscribed in the circle, the diameter of the circle is equal to the diagonal of the square.

2r = s√2
Squaring both sides, we get:
4r² = 2s²or s² = 2r²
Dividing by 2 on both sides, we get:
s²/2 = r²
Differentiating both sides with respect to t, we get:
ds²/dt = 2r (dr/dt)
Dividing both sides by 2s, we get:
ds/dt = r (dr/dt) / s

Substituting r² = s²/2,

[tex]ds/dt = r (dr/dt) / √2s2s ds/dt = r (dr/dt)s ds/dt = (r/2) (dr/dt)2s ds/dt = r (dr/dt) or dA/dt = 2s ds/dt = 2r (dr/dt)[/tex]

Now, substituting r² = s²/2,

dA/dt = 2s ds/dt = 2(√2 s) (dr/dt) = 2(√2) r (dr/dt)

Now, substituting dr/dt = 1 in/min and r = 6 in (since the circle is increasing at 1 in/min, the radius after t minutes is 6 + t),

dA/dt = 2(√2) (6 + t) (1) = 12√2 + 2√2t square inches/minute

Therefore, the area of the square is changing at a rate of 12√2 + 2√2t square inches/minute.

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No, it is not less likely that you will get one of the four types of juice, as they are the only options available.

Probability of getting apple juice = 29%

                                                      = 0.29

probability of getting grape juice = 31%

                                                       = 0.31

Probability of getting orange juice = 19%

                                                      = 0.19

Probability of getting pear juice = 21%

                                                      = 0.21

The total probabilities of getting apple, grape, orange, and pear juice sum up to

= 29% + 31% + 19% + 21%

= 100%.

or

= 0.29 + 0.31 + 0.19 + 0.21

= 1.00

This implies,

That every customer will get one of the four types of juice, and there is a 100% chance of getting one of them.

Therefore, the given probabilities are not less likely as it represents chances of one type of juice definitely will be given.

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The vertical axis of the graph represents profit, so the x-intercepts represent prices in the produce 0 profit. The maximum value of the graph is the maximum profit that can be obtained for anyprice

The higher or largest number of the chart is the maximum reach

B) We read the value of f(1) from the graph to be about 120, so the average rate of change is about:

(f(4) -f(1))/(4 -1) = (270 -120)/(3) = 50

The average rate of the change from x = 1 to x = 4 is about 50.* This means profit will increase on average $50 for each $1 increase in price in what interval.

If we take the peak profit to be $270 we can write f(x) as:

f(x) 16.875x(x-8)

Then f(1) = 118.15 and average rate of change is 50.625.

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