A group of students formed a circle during a game.The circumference of the circle was about 43.96 feet,and the diameter of the circle was 14 feet.Which expression best represents the value of x?

Answer:

a) The demand function is

[tex]q(p) = -4 p + 107[/tex]

b) The nightly revenue is

[tex] R(p) = -4 p^2 + 107 p [/tex]

c) The profit function is

[tex]P(p) = -4 p^2 + 133.75 p - 939 [/tex]

d) The entrance fees that allow Swing Haven to break even are between 10.03 and 23.41 dollars per guest.

Step-by-step explanation:

a) Lets find the slope s of the demand:

[tex] s = \frac{79-43}{7-16} = \frac{36}{-9} = -4 [/tex]

Since the demand takes the value 79 in 7, then

[tex]q(p) = -4 (p-7) + 79 = -4 p + 107[/tex]

b) The nightly revenue can be found by multiplying q by p

[tex]R(p) = p*q(p) = p*( -4 p + 107) = -4 p^2 + 107 p[/tex]

c) The profit function is obtained from substracting the const function C(p) from the revenue function R(p)

[tex]P(p) = R(p) - C(p) = p*q(p) = -4 p^2 + 107 p - (-26.75p + 939) = \\\\-4 p^2 + 133.75 p - 939[/tex]

d) Lets find out the zeros and positive interval of P. Since P is a quadratic function with negative main coefficient, then it should have a maximum at the vertex, and between the roots (if any), the function should be positive. Therefore, we just need to find the zeros of P

[tex]r_1, r_2 = \frac{-133.75 \,^+_-\, \sqrt{133.75^2-4*(-4)*(-939)} }{-8} = \frac{-133.75 \,^+_-\, 53.526}{-8} \\r_1 = 10.03\\r_2 = 23.41[/tex]

Therefore, the entrance fees that allow Swing Haven to break even are between 10.03 and 23.41 dollars per guest.

We found the linear demand equation as q(p) = -4p + 107. The nightly revenue R(p) is R(p) = -4p^2 + 107p, and the profit P(p) is P(p) = -4p^2 + 133.75p - 939. Setting the profit equal to zero, we found the club breaks even at entrance fees $5.32 and $44.31.

In this question, we're asked to formulate linear demand, revenue, and profit equations, and find the entrance fees for break even. First, let's find the linear demand equation.

To obtain this demand equation, we can use the two given points ($7, 79) and ($16, 43) and find the slope (rate of change) equals (43-79) / (16-7) = -4. Hence the demand equation will be q(p) = -4p + b. To find b, substitute one of the points into the equation, for example ($7, 79), we get b=107, hence q(p) = -4p + 107.

Next, let's find the nightly revenue R (p), which is simply the product of the entrance fee and the number of guests, hence R(p) = p * q(p) = p (-4p + 107) = -4p^2 + 107p.

 The profit P(p) is the difference between revenue and costs, hence P(p) = R(p) - C(p) = -4p^2 + 107p - (-26.75p + 939) = -4p^2 + 133.75p - 939.

Finally, for Swing Haven to break even, the profit must be zero. By setting P(p) = 0 and solving the equation -4p^2 + 133.75p - 939 = 0, we get the two solutions p = 5.3216 and p = 44.3031, or roughly $5.32 and $44.31.

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Answer:

D) 8 inches

Step-by-step explanation:

Since the ratio is 2:60 inches, therefore the model should be:

2/60 X 240=8 inches tall, or 8/12=2/3 foot tall.

Answer:

40/13 hours

Step-by-step explanation:

Let 1 represent the room being painted.

1/5 would represent 1/5 of a room painted in 1 hour.

1/8 would represent 1/8 of a room being painted in 1 hour

But they are are working together so

(1/5)x + (1/8)x = 1

The lowest common denominator is 5 * 8 = 40

[(1/5)x * 40 + (1/8)x * 40] = 1

8x + 5x = 40

13x = 40

x = 40/13

x = 3.077 hours

But you want a fraction as your answer so it is 3 1/13 or 40/13

You should notice that 3.077 is smaller than the lowest amount of time both of them use. That always happens with these questions.

Final answer:

Rachel and Barbara would take ⅜shy hours to paint the living room together, as they have a combined work rate of 0.325 room/hour calculated from their rates of 1 room per 5 hours and 1 room per 8 hours, respectively.

Explanation:

To find out how many hours it would take Rachel and Barbara to paint the living room together, we can use the concept of work rate. Rachel's work rate is ⅑or (1 room per 5 hours), and Barbara's work rate is ⅑or (1 room per 8 hours). To find their combined work rate, we add their rates together.

Rachel's rate: ⅑or (1 room/5 hours) = 0.2 room/hour
Barbara's rate: ⅑or (1 room/8 hours) = 0.125 room/hour
Combined rate: 0.2 + 0.125 = 0.325 room/hour

To find out how long it takes them to paint the room together, we can set up the equation: 1 room = (0.325 room/hour) × (hours). Solving for 'hours' gives us:

Hours = ⅑or / 0.325
Hours = ⅑or / (⅜shy / 1)
Hours = ⅜shy × 1
Hours = ⅜shy
Therefore, it would take them ⅜shy hours to complete the painting together.

Answer:

C. [tex]-0.00005d^2+11.8d-9000[/tex]

Step-by-step explanation:

Given:

Cost Price for producing 'd' dog lashes = [tex]9000+3.2d[/tex]

Revenue Generated from selling 'd' dog lashes = [tex]d(15-0.00005d)[/tex]

We need to find the profit earned from producing and selling 'd' dog lashes.

Solution:

Now we know that;

profit earned from producing and selling 'd' dog lashes can be calculated by Subtracting Cost Price for producing 'd' dog lashes from Revenue Generated from selling 'd' dog lashes.

framing in equation form we get;

Profit earned = [tex]d(15-0.00005d)-(9000+3.2d)[/tex]

Now Applying Distributive property we get;

Profit earned = [tex]15d-0.00005d^2-9000-3.2d[/tex]

Now Combining like terms we get;

Profit earned = [tex]-0.00005d^2+15d-3.2d-9000[/tex]

Profit earned = [tex]-0.00005d^2+11.8d-9000[/tex]

Hence  Profit earned from producing and selling 'd' dog lashes is [tex]-0.00005d^2+11.8d-9000[/tex].

Final answer:

The polynomial representing the profit from producing and selling d dog leashes is -0.00005d² + 11.8d - 9000, calculated by subtracting the cost (9000 + 3.2d) from the revenue (d(15 - 0.00005d)).

Explanation:

The student's question relates to finding the polynomial expression that represents the profit earned from producing and selling d dog leashes. Profit is calculated by subtracting the cost from the revenue. The cost polynomial is given as 9000 + 3.2d, and the revenue polynomial is d(15 - 0.00005d).

To find the profit, we subtract the cost from the revenue:

Profit = Revenue - Cost

Profit = (d(15 - 0.00005d)) - (9000 + 3.2d)

Profit = 15d - 0.00005d² - 9000 - 3.2d

Profit = -0.00005d² + (15 - 3.2)d - 9000

Profit = -0.00005d² + 11.8d - 9000

Therefore, the correct polynomial that represents the profit is -0.00005d² + 11.8d - 9000.

Part 1: The function that models the height of the basketball is [tex]\( h(t) = -16t^2 + 15t + 6.5 \).[/tex]

Part 2: The basketball hits the ground after approximately 0.97 seconds.

Part 3: The basketball reaches its maximum height at approximately 0.47 seconds.

Part 4: The maximum height of the basketball is approximately 11.39 feet.

Part 1: To model the height of the basketball, we use the given function [tex]\( h(t) = -16t^2 + v0t + h0 \)[/tex] with the initial velocity [tex]\( v0 = 15 \)[/tex] ft/sec and the initial height [tex]\( h0 = 6.5 \)[/tex] feet. Plugging in these values, we get the function:

[tex]\[ h(t) = -16t^2 + 15t + 6.5 \][/tex]

Part 2: To find the time it takes for the basketball to hit the ground, we set the height function equal to zero and solve for [tex]\( t \)[/tex]:

[tex]\[ -16t^2 + 15t + 6.5 = 0 \][/tex]

Using the quadratic formula [tex]\( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \),[/tex]where [tex]\( a = -16 \), \( b = 15 \)[/tex], and [tex]\( c = 6.5 \)[/tex], we get two solutions. We discard the negative solution because time cannot be negative, and we round the positive solution to the nearest hundredth:

[tex]\[ t = \frac{-15 \pm \sqrt{15^2 - 4(-16)(6.5)}}{2(-16)} \] \[ t = \frac{-15 \pm \sqrt{225 + 416}}{-32} \] \[ t = \frac{-15 \pm \sqrt{641}}{-32} \] \[ t \approx \frac{-15 + 25.31}{-32} \] \[ t \approx 0.97 \text{ seconds} \][/tex]

Part 3: To find when the basketball reaches its maximum height, we need to find the vertex of the parabola. The time coordinate of the vertex of a parabola [tex]\( ax^2 + bx + c \)[/tex] is given by [tex]\( t = -\frac{b}{2a} \)[/tex]. For our function, [tex]\( a = -16 \)[/tex]and [tex]\( b = 15 \)[/tex], so:

[tex]\[ t = -\frac{15}{2(-16)} \] \[ t = \frac{15}{32} \] \[ t \approx 0.47 \text{ seconds} \][/tex]

Part 4: To find the maximum height, we substitute the time at which the maximum height is reached back into the height function:

[tex]\[ h(0.47) = -16(0.47)^2 + 15(0.47) + 6.5 \] \[ h(0.47) \approx -16(0.2209) + 7.05 + 6.5 \] \[ h(0.47) \approx -3.5344 + 7.05 + 6.5 \] \[ h(0.47) \approx 11.39 \text{ feet} \][/tex]

Therefore, the basketball reaches a maximum height of approximately 11.39 feet after approximately 0.47 seconds.

Answer:

[tex]k=250[/tex]

Step-by-step explanation:

If y varies inversely with x, then

[tex]y=\dfrac{k}{x},[/tex]

where k is the constant of variation or the constant of inverse proportionality.

Since [tex]x=2.5[/tex] when [tex]y=100,[/tex] you have that

[tex]100=\dfrac{k}{2.5}[/tex]

Evaluate k:

[tex]k=100\cdot 2.5\\ \\k=250[/tex]

and

[tex]y=\dfrac{250}{x}[/tex]

Answer:

Attached is the image of the solution . cheers

Final answer:

To solve for the distance, velocity, and acceleration of a hammer dropped from a hovering helicopter, calculations based on the given formula s(t) = 1818t^2 are used, yielding a fall of 29128 feet in 4 seconds, a velocity of 14512 feet/sec at 4 seconds, and a constant acceleration of 3636 feet/sec^2.

Explanation:

When an object is dropped on a certain earth-like planet, the distance it falls in t seconds, assuming that air resistance is negligible, is given by s(t) = 1818t2, where s(t) is in feet. To solve the problem involving a medic's reflex hammer dropped from a hovering helicopter:

(a) To find how far the hammer falls in 4 seconds, substitute t = 4 into the equation: s(4) = 1818(4)2 = 29128 feet.

(b) The velocity of the hammer after 4 seconds can be found using the derivative of s(t), v(t) = 2×1818×t. Substituting t = 4, v(4) = 2×1818×4 = 14512 feet/sec.

(c) The acceleration of the hammer is constant and equal to 2×1818 feet/sec2 = 3636 feet/sec2, which is twice the coefficient in the equation for s(t).

These calculations demonstrate the principles of kinematics, specifically how position, velocity, and acceleration relate to one another for an object in free fall on an earth-like planet with negligible air resistance.

Answer:

8/9 of a cup

Step-by-step explanation:

8 cups of icecream to be divided among 9 people, so they can't have full cups.

Each person will get 8/9 of a cup

Answer:

Probability p( selecting 8 cities alphabetically) = 2.48×10^-5

Step-by-step explanation:

Number of possible ways of choosing 8 cities=Permutation P(n,k) = n!/(n-k)!

P(8,8)= 8!/(8-8)! = 8! = 40,320

Probability (selecting 8 cities alphabetically) = 1/40320 = 2.48×10^-5

Answer:

The answer is 0.00002480

Step-by-step explanation:

From the question stated let us recall the following statement.

The number of cities the tourist wants to visit is =8

Now,

If the route is selected randomly, what is the probability that the cities she visits are in alphabetical order.

Therefore,

The probability that she visits the cities in alphabetical order = 1/8!

There are 8! = 40320 ways in which these cities can be visited

P(Visiting in alphabetical order) = 1/40320 = 0.00002480

Answer:

Step-by-step explanation:

Triangle WXY is a right angle triangle.

From the given right angle triangle,

WX represents the hypotenuse of the right angle triangle.

With m∠W as the reference angle,

WY represents the adjacent side of the right angle triangle.

XY represents the opposite side of the right angle triangle.

To determine m∠W, we would apply

the cosine trigonometric ratio.

Cos θ = adjacent side/hypotenuse. Therefore,

Cos W = 5/7 = 0.7143

W = Cos^-1(0.7143)

W = 44.1° to the nearest tenth.

Answer: He makes 23 pints this year .

Step-by-step explanation:

Given : Oliver makes blueberry jam every year the number of pints of jam he makes this can be represented by the expression

[tex]4p - 9[/tex] , where  p = the number of pints of jam he made last year.

if Oliver made 8 pints of jam last year , then p = 8

Substitute value p= 8 in the given expression  , we get

Then, the number of  pints he make this year = [tex]4(8)-9 = 32-9=23[/tex]

Hence, the number of pints Oliver make this year = 23

Answer:

The probability that Meena wins is 21/25

Step-by-step explanation:

In order for Meena to win, she needs to win the next turn or the following one, otherwise, she loses. The probability for that is equal to substract from 1 the probability of the complementary event: bob wins in the next 2 turns. Since each turn is independent from the other, we can obtain the probability of Bob winning the next 2 turns by taking the square of the probability of him winning on one turn, hence it is

[tex] {\frac{2}{5}}^2 = \frac{4}{25} [/tex]

Thus, the probability for Meena to win is 1-4/25 = 21/25.

Meena only needs one more point to win, and with a ⅘ or 60% probability in her favor for the next turn, she will win the game.

The question asks for the probability that Meena will win a game where she is currently ahead 9 to 6, and the game is won by the first person to reach 10 points. In each turn, there is a ⅗ chance that Bob will get 2 points and Meena will lose 1 point, and a ⅘ chance that Meena will get 2 points and Bob will lose 1 point.

To find the probability of Meena winning, we only need to look at the next round because Meena is already at 9 points and she needs only 1 point to win. Two scenarios can occur:

Bob gets 2 points, and Meena loses 1 point. This outcome is not possible because Meena cannot have 8 points; she's already at 9 points. So, this situation doesn't count.

Meena gets 2 points (which will put her at or above 10 points) and Bob loses 1 point. This is the winning scenario for Meena. The probability of this happening is ⅘ or 0.6 (since only this outcome will end the game with Meena as the winner).

Therefore, the probability that Meena will win on the next turn (and win the game) is 0.6 or 60%

Answer:

x = 0 or -4

Step-by-step explanation:

x² + 4x = 0

To complete the square, take half of the middle coefficient, square it, then add to both sides.

(4/2)² = 2² = 4

x² + 4x + 4 = 4

(x + 2)² = 4

x + 2 = ±2

x = -2 ± 2

x = 0 or -4

The differential equation provided is transformed to M(x,y) dx + N(x,y) dy = 0 with M(x,y) = [tex]-4x^4 - 3y[/tex]and N(x,y) = 3x + [tex]2y^4[/tex], setting the stage for verifying its exactness through partial differentiation.

To address the student's query about finding a function F(x,y) whose differential fits the given differential equation, we'll first transform the differential equation dy/dx =[tex](-4x4-3y)/(3x+2y4)[/tex] into the form M(x,y) dx + N(x,y) dy = 0.

This transformation requires us to regard the equation as a differential one, implying:

This representation simplifies the process of checking for exactness, which necessitates partial differentiation and comparison of ∂M/∂y and ∂N/∂x.

Should these partial derivatives be equal, the differential is exact, enabling the determination of the function F(x, y) through integration.

Answer:

5 inches

Step-by-step explanation:

See attachment for explanation.

Answer:

The submarine will be located 1350ft below sea level or (-1350ft)

Step-by-step explanation:

First we must find the distance traveled

[tex](\frac{300}{1} )(\frac{4.5}{1})[/tex]

From this, we are able to calculate that traveled distance of the submarine.

300 x 4.5 = 1350

1350 + 0 (sea level) = 1350

Therefore the distance traveled by the submarine is 1350ft

Answer:

-1350 ft

Step-by-step explanation:

Answer:

The answer to your question is dAC = 10 units

Step-by-step explanation:

Data

From the graph, take the coordinates of the points A and C.

A (3, - 1)

C (-5, 5)

Process

Use the formula of the distance between two points to find the length

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

x1 = 3    y1 = -1

x2 = -5  y2 = 5

Substitution

dAC = [tex]\sqrt{(-5 - 3)^{2} + (5 + 1)^{2}}[/tex]

Simplification

dAC = [tex]\sqrt{(-8)^{2} + (6)^{2}}[/tex]

dAC = [tex]\sqrt{64 + 36}[/tex]

dAC = [tex]\sqrt{100}[/tex]

dAC = 10 units

Answer:

The height of the building is 153.664 meter.

Step-by-step explanation:

Consider the provided function.

[tex]s(t) = 4.9t^2 + v_0 t + s_0[/tex]

Here t represents the time v₀ represents the initial velocity, s₀ represents the initial height and s(t) represents the height after t seconds.

It is given that the splash is seen 5.6 seconds after the stone is dropped.

That means after 5.6 seconds height s(t) = 0, Also the initial velocity of the stone is 0.

Substitute respective values in the above function.

[tex]0 = 4.9(5.6)^2 +5.6(0)+ s_0[/tex]

[tex]0 = 4.9(31.36)+ s_0[/tex]

[tex]s_0=-153.664[/tex]

As height can't be a negative number so the value of s₀ is 153.664.

Hence, the height of the building is 153.664 meter.

Answer:

calculator that what I got 56

Step-by-step explanation:

Answer: [tex]x=5[/tex]

Step-by-step explanation:

The area of a rectangle can be found with the following formula:

[tex]A=lw[/tex]

Where "l" is the length and "w" is the width.

In this case you can identify in the figure given in the exercise that:

[tex]l=4x-2\\\\w=3[/tex]

You know that the area of that rectangle is the following:

[tex]A=54[/tex]

Therefore, knowing those values, you can substitute them into the formula and then you must solve for "x" in order to find its value. You get that this is:

[tex]54=(4x-2)(3)\\\\54=12x-6\\\\54+6=12x\\\\\frac{60}{12}=x\\\\x=5[/tex]

Answer: [tex](x+6)^{2}+(y+5)^{2}=r^{2}[/tex]

Step-by-step explanation:

The formula for finding the equation of circle with center (a,b) is given as :

[tex](x-a)^{2}+(y-b)^{2}=r^{2}[/tex]

The end point of the diameter is given as :

(-10, -6) and (-2, -4) , this means that the coordinate of the center is the Mid -point of the end point .

The mid - point = ( -6 , - 5)

substituting into the formula , we have

[tex](x-(-6))^{2}+(y-(-5))^{2}[/tex][tex]= r^{2}[/tex]

[tex](x+6)^{2}+(y+5)^{2}=r^{2}[/tex]

This is the equation of the circle

Answer:

Vertical compression, horizontal shift left, vertical shift down

Step-by-step explanation:

[tex]f(x) = 3^x => g(x) = \frac{1}{2} *2^{x+3} -5[/tex]

Let's break down what happened here:

[tex]\frac{1}{2}[/tex]  - This indicates a vertical compression

[tex]2^{x+3}\\[/tex]   - This indicates a horizontal shift left

-5    - This indicates a vertical shift down

Final answer:

Transforming f(x) to g(x) involves a vertical compression by ½, a horizontal shift left by 3 units, and a vertical shift down by 5 units. There is also an exponential base change, but it does not correspond directly to the given transformation options.

Explanation:

To transform the function f(x) = 3x into g(x) = ½ · 2x+3 - 5, let's analyze each part of the transformation. Break down g(x) into its components to determine the transformations:

Based on this analysis, the correct transformations that occurred are a vertical compression, a horizontal shift left, and a vertical shift down.

Answer: b. 70%

Step-by-step explanation:

Given : Nik logs the following hours meeting with clients (c) or doing other work (o) :

Mon: 6 c, 4 o

Tue: 8 c, 2 o

Wed: 9 c, 1 o

Thu: 7 c, 3 o

Fri: Off

The number of hours Nik spend with clients on Thursday = 7 [Number of corresponding to c is 7 in the table]

Total hours he spend in work on Thursday =  7+3 = 10

The percent of time Nik spent with clients on Thursday :

[tex]\dfrac{\text{Number of hours he spent with clients}}{\text{Total works he work on Thursday}}\times100\\\\=\dfrac{7}{10}\times100=7\times10\%=70\%[/tex]

Hence, the Nik spent 70% of his time with clients on Thursday.

Thus , the correct option is b. 70%.

Combining 2s and −4s gives an equivalent expression of −2s.

We need to combine the like terms. Like terms are terms that have the same variable raised to the same power. In this case, both terms involve the variable "s."

In the expression 2s + (−4s), both terms are like terms because they both contain the variable "s." The coefficients are 2 and −4.

To combine like terms, you simply add or subtract their coefficients while keeping the variable the same:

Coefficient of the first term: 2

Coefficient of the second term: −4

Now, perform the arithmetic operation:

2 + (−4) = −2

Answer:

option 1

Step-by-step explanation:

first we have to find the slopes of the lines

D(1, -2)     E(3, 4)

y = m*x + b

m1: slope

m1 = (y2-y1) / (x2-x1)

m1 = (4 - (-2)) / (3 - 1)

m1 = 4+2 / 3-1

m1 = 6 / 2

m1 = 3

we do the same with the other 2 points

D(-1, 2)     E(4, 0)

y = m*x + b

m2: slope

m2 = (y2-y1) / (x2-x1)

m2 = (0 - 2) / (4 - (-1))

m2 = -2 / 4 + 1

m2 = -2 / 5

m1 = 3        m2 = -2/5

for 2 lines to be perpendicular it must be met

m1 * m2 = -1

we check if they are perpendicular

3 * -2/5 = -1

-6/5 = -1   <-- no perpendicular

Two pounds of gummy bears will cost $2.67. The unit rate for each pounds of gummy bears is measured in dollars.

Word problems in mathematics involve the use of mathematical concepts and arithmetic operations to solve real-life cases. It involves a careful understanding of the problem you want to solve.

From the parameters given:

By cross multiplying, we have:

[tex]\mathbf{x = \dfrac{2 \ pounds \times \$4.00}{\$3.00}}[/tex]

x = $2.67

Learn more about word problems in mathematics here:

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Final answer:

The cost for two pounds of gummy bunnies is $2.66, with the unit rate being $1.33 per pound after rounding to two decimal places.

Explanation:

To determine how much two pounds of gummy bunnies will cost at Joan's candy emporium, we first need to calculate the unit rate of the gummy bunnies that are selling for $4.00 per three pounds. The unit rate is found by dividing the total cost by the number of pounds:

Unit Rate = Total Cost / Number of Pounds

Unit Rate = $4.00 / 3 pounds = $1.33 per pound (rounded to two decimal places)

Now that we have the unit rate, we can determine the cost for two pounds of gummy bunnies:

Cost for Two Pounds = Unit Rate x Number of Pounds

Cost for Two Pounds = $1.33 per pound x 2 pounds = $2.66 (rounded to two decimal places)

Answer:

200

Step-by-step explanation:

50 doubles to 100 and then doubles again to 200.

Answer:

(a) 0.0024

(b) 0.0012

(c) 0.0006

(d) 0.0004

Step-by-step explanation:

The total number of possible integers when any number is selected is 10 (i.e from 0 - 9). When four number integers are selected, the total number of sample sample will be;

                                   10 × 10 × 10 × 10 = 10,000

The sample space = 10,000

To know the possible ways of selecting the given four digits, we will use permutation.

                                 [tex]^{n}P_{r} = \frac{n!}{(n-r)!}[/tex]

To get the probability,

[tex]Probability \ of \ winning (Selected \ numbers) = \frac{number\ of\ possible\ outcomes\ of\ selected\ numbers}{sample\ space}[/tex]

(a) When 6,7,8,9 are selected, n = 4 , r = 4

The possible ways of selecting 6,7,8,9 is;

                                    [tex]^{4}P_{4} = \frac{4!}{(4-4)!}[/tex]

                                            [tex]= \frac{4!}{(0)!}[/tex]

but 0! = 1

                                     [tex]^{4}P_{4} = 4![/tex]

                                     = 4 × 3 × 2 × 1 = 24

                 [tex]Prob (6,7,8,9) = \frac{24}{10000} = 0.0024[/tex]

(b) When 6, 7, 8, 8 are selected,

The possible ways of selecting 6,7,8,8 is;

                                     [tex]= \frac{4!}{1! \ 1! \ 2!}[/tex]

                                     [tex]= \frac{4!}{2!}[/tex]  

                                     [tex]=\frac{4 * 3 * 2 * 1}{2 * 1}[/tex]

                                     = 12

             [tex]Prob (6,7,8,8) = \frac{12}{10000} = 0.0012[/tex]

(c) When 7, 7, 8, 8 are selected,

The possible ways of selecting 7,7,8,8 is;

                                     [tex]= \frac{4!}{2! \ 2!}[/tex]

                                     [tex]=\frac{4 * 3 * 2 * 1}{(2 * 1)(2 * 1)}[/tex]

                                     = 6

             [tex]Prob (7,7,8,8) = \frac{6}{10000} = 0.0006[/tex]

(d) When 7, 8, 8, 8 are selected,

The possible ways of selecting 7,8,8,8 is;

                                    [tex]= \frac{4!}{1! \ 3!}[/tex]    

                                    [tex]=\frac{4 * 3 * 2 * 1}{3 * 2 * 1}[/tex]    

                                    = 4

             [tex]Prob (7,8,8,8) = \frac{4}{10000} = 0.0004[/tex]