Given \( f(x)=5 x \) and \( g(x)=7 x^{2}+6 \), find the following expressions. (a) \( (f \circ g)(4) \) (b) \( (g \circ f)(2) \) (c) \( (f \circ f)(1) \) (d) \( (g \circ g)(0) \) (a) (fog)(4) = (Simplify your answer.)
(b) (go f)(2) = (Simplify your answer.)
(c) (fo f)(1) = (Simplify your answer.)
(d) (go g)(0) = (Simplify your answer.)

Based on the graph of this linear function, the domain and range are as follows;

Domain = {0, 100}.

Range = {450, 1200}.

In Mathematics, a domain is the set of all real numbers for which a particular function is defined.

Additionally, the vertical extent of any graph of a function represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.

By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:

Domain = {0, 100}.

Range = {450, 1200}.

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Answer:

C. 10 sandwiches

Step-by-step explanation:

Each section of a box plot makes up 25% of it. Since Q1 is marked by $18, that means 25% of 40 sandwiches can be bought with it. So, [tex]\frac{40}{4}[/tex] = 10.

The question involves a basic division operation in Mathematics. Without the cost of a sandwich, it's impossible to answer. If provided, just divide the total money by the price of a sandwich to find the number that can be purchased.

The student's question involves simple division, which is a topic under the subject of Mathematics. To determine how many sandwiches one can afford with $18, it is necessary to have the cost of one sandwich. Unfortunately, without the given price of a single sandwich as indicated in the boxplot, we cannot compute the answer.

But if we are provided the cost of a single sandwich, we would simply divide the total amount of money ($18) by the cost of a single sandwich. This would give us the total number of sandwiches affordable with the given amount of money.

Assuming a sandwich costs $2, here is how the computation would be: $18 ÷ $2 = 9 sandwiches. So, with $18, we could afford 9 sandwiches at $2 each.

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The probability distribution function of Y = |X| is given by:

P(Y ≤ y) = P(|X| ≤ y) = P(-y ≤ X ≤ y)

Since X ~ N(0, 2), we can use the standard normal distribution to find the probability;

P(Y ≤ y) = P(Z ≤ y/√2) - P(Z ≤ -y/√2)

where Z is a standard normal random variable.

To find E(|X|), we can use the formula:

E(|X|) = √(2/π) * σ

where σ is the standard deviation of X. Since X ~ N(0, 2), σ = √2, so:

E(|X|) = √(2/π) * √2 = √(4/π)

To find Var(|X|), we can use the formula:

Var(|X|) = E(X^2) - E(|X|)^2

Since X ~ N(0, 2), E(X^2) = σ^2 = 2. And we already found E(|X|) = √(4/π), so:

Var(|X|) = 2 - (4/π) = (2π - 4)/π

Therefore, the probability distribution function of Y = |X| is P(Y ≤ y) = P(Z ≤ y/√2) - P(Z ≤ -y/√2), the expected value of |X| is E(|X|) = √(4/π), and the variance of |X| is Var(|X|) = (2π - 4)/π.

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The answer is 6a+5.

Distribute the negative sign to the second polynomial:
(9a+6)-(3a+1) = (9a+6)+(-3a-1)

Combine like terms:
(9a+6)+(-3a-1) = (9a-3a)+(6-1) = 6a+5

Therefore, the answer is 6a+5.

So, the subtraction of the polynomials (9a+6)-(3a+1) is 6a+5.

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[tex]\textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \hspace{5em}\stackrel{center}{(\underset{-3}{h}~~,~~\underset{6}{k})}\qquad \stackrel{radius}{\underset{5}{r}} \\\\[-0.35em] ~\dotfill\\\\ ( ~~ x - (-3) ~~ )^2 ~~ + ~~ ( ~~ y-6 ~~ )^2~~ = ~~5^2\implies (x+3)^2+(y-6)=25[/tex]

Answer: See attached.

Step-by-step explanation:

     First, we will find the unit rate by dividing.

312 miles / 8 gallons = 39 miles per gallon

     Then, we can fill out the other values using this unit rate.

39 miles / 39 miles per gallon = 1 gallon

3 gallons * 39 miles per gallon = 117 miles

       Lastly, we will use these two equivalent ratios we found above to fill out the table and then plot the points. See attached.

Answer:

429 m³

Step-by-step explanation:

The volume of right rectangular = Length x Width x Height

The volume of right rectangular prism = 12 x 5 1/2 x 6 1/2

                                                                      = 12 x 11/2 x 13/2

                                                                      = 3 x 11 x 13

                                                                      = 429 m³

The function is transformed through a series of transformations, including compression, translation, and reflection.

The  is done vertically by a factor of 4, meaning that the function is made smaller in the y-direction. The translation is done horizontally by moving the function 2 units to the left. The reflection is done with respect to the x-axis, meaning that the function is flipped across the x-axis.

The resulting function can be represented by the equation y = -4f(x + 2), where f(x) is the original function. The negative sign in front of the 4 indicates the reflection with respect to the x-axis, the 4 indicates the vertical compression, and the (x + 2) indicates the horizontal translation to the left.

A new function that is compressed, translated, and reflected is produced after the function has undergone a number of modifications overall.

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Using the sine rule and triangle ABC,   the height  is 14.43375ft. BAC=40° and ABC=30°

if C is 20° and  A is 50°

ABC=50°-20°=30°

BCA=20°+90°=110°

ABC+BCA+BAC=180°

30°+110°+BAC=180°

BAC=180°-140°=40°

Using the sine rule and triangle ABC,

opposite=x

adjacent =25 m.

tanθ =x/25

Multiplying both sides by 25 gives

x=25* tan 30° =25* 0.57735.=14.43375

To put it another way, there is only one plane that contains all of the triangles. All triangles are enclosed in a single plane on the Euclidean plane, however this is no longer true in higher-dimensional Euclidean spaces. Unless otherwise specified, this article deals with triangles in Euclidean geometry, specifically the Euclidean plane.

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The solution of the given system of the equation will be (0, 1), and (4, 9).

Simultaneous equations, a system of equations Two or more equations in algebra must be solved jointly (i.e., the solution must satisfy all the equations in the system). The number of equations must match the number of unknowns for a system to have a singular solution.

There are four methods for solving systems of equations: graphing, substitution, elimination, and matrices.

Given a system of equations such that,

y = x² - 2x +1

y = 2x + 1

By subtracting both equations we will get,

x²-2x +1 - 2x -1 = 0

x² -4x = 0

x = 0, x = 4

at this value

y = 1, y = 9

therefore, the solution of the given system of the equation will be (0, 1), and (4, 9).

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If the divisor contains 2 or more terms, (6x^(2)y+18x^(2)y^(2)-xy^(2))/(6xy) it can simplifies to x+3xy-y.

To divide the given expression, we need to factor out the common term from the numerator and then simplify by canceling out the common terms from the numerator and denominator. Here is the step-by-step explanation:

Step 1: Factor out the common term from the numerator:
(6x²y+18x²y²)-xy²)/(6xy) = 6xy(x+3xy-y)/(6xy)

Step 2: Cancel out the common terms from the numerator and denominator:
6xy(x+3xy-y)/(6xy) = (x+3xy-y)

Step 3: Simplify the expression:
(x+3xy-y) = x+3xy-y

Therefore, the final answer is x+3xy-y.

So, the given expression (6x²y+18x²y²-xy²)/(6xy) simplifies to x+3xy-y.

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We are only interested in the critical numbers on the interval [1,3], so we can disregard x = 0. Therefore, the critical numbers of the function f(x) on the interval [1,3] are x = 3.

The critical numbers of a function are the points where the derivative of the function is either zero or undefined. To find the critical numbers of the given function f(x) = x^2/(3-x), we need to first find its derivative:

f'(x) = (2x(3-x) - (-1)x^2)/ (3-x)^2 = (6x - x^2 - x^2)/ (3-x)^2 = (6x - 2x^2)/ (3-x)^2

Now, we need to find the values of x for which f'(x) = 0 or f'(x) is undefined. f'(x) is undefined when the denominator (3-x)^2 is equal to 0, which occurs when x = 3. f'(x) is equal to 0 when the numerator 6x - 2x^2 is equal to 0:

6x - 2x^2 = 0
2x(3 - x) = 0
x = 0 or x = 3

However, we are only interested in the critical numbers on the interval [1,3], so we can disregard x = 0. Therefore, the critical numbers of the function f(x) on the interval [1,3] are x = 3.

Answer: 3

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After evaluating using the correct order of operations, the result will become 22/9.

The correct order of operations is to perform any calculations inside parentheses first, then exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right. This order of operations is known as PEMDAS. Using this order, we can evaluate the expression as follows:

(17)/(18) + (6)/(35) ÷ (5)/(7) x (25)/(4)

Divide 6/35 by 5/7 by multiplying 6/35 by the reciprocal of 5/7.
= (17/18) + (6/35) x (7/5) x (25/4)
= (17/18) + (6/25) x (25/4)

Multiply 6/25 by 25/4.
= (17/18) + (6/4)

Finally, add 17/18 and 6/4 together.
= 22/9

Therefore, the final answer is 22/9,

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To get the chart that you have circled in the second picture, you need to create a table using the given formulas and then create a chart using the table. Here are the steps:

1. Create a table with the following columns: Iteration, Random, Demand, Revenue, Cost, Refund, and Profit.
2. In the first row of the table, enter the given formulas in the respective columns. For example, in the first row of the Random column, enter =RAND(), in the first row of the Demand column, enter =ROUND ( NORM.INV ( RAND(), Mean, Standard Deviation),0), and so on.
3. Copy the formulas down to the 5000th row to get the values for all 5000 iterations.
4. Select the entire table and click on the Insert tab in the Excel ribbon.
5. In the Charts group, click on the type of chart that you want to create. In this case, it looks like you want to create a scatter chart.
6. In the Chart Design tab, click on the Select Data button in the Data group.
7. In the Select Data Source dialog box, click on the Add button in the Legend Entries (Series) section.
8. In the Edit Series dialog box, enter a name for the series, select the Profit column for the Series X values, and select the Demand column for the Series Y values.
9. Click on the OK button to close the Edit Series dialog box and then click on the OK button to close the Select Data Source dialog box.
10. Your chart should now be created and should look like the one that you have circled in the second picture.

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Using composition function , the price of the computer after both discounts are applied is 0.7x - 140.

To find the price of the computer after both discounts are applied, we can use composition function . Specifically, we can find the composition of f and g, denoted as f(g(x)) or (f o g)(x). This will give us the price of the computer after both discounts are applied.

To find f(g(x)), we can substitute the expression for g(x) into the function f(x):

f(g(x)) = f(0.7x) = (0.7x) - 200

So the price of the computer after both discounts are applied is (0.7x) - 200.

Alternatively, we could find the composition of g and f, denoted as g(f(x)) or (g o f)(x). This will give us the same result, since the order of the discounts does not matter.

To find g(f(x)), we can substitute the expression for f(x) into the function g(x):

g(f(x)) = g(x-200) = 0.7(x-200) = 0.7x - 140


Either way, we can see that the price of the computer after both discounts are applied is a function of the original price x, and can be represented by the composition of the functions f and g.

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Answer:

The answer to your problem is, 9

Step-by-step explanation:

I can conclude that the answer is nine by the following step shown down below.

The bottom triangle the one that it on the " floor ".

It is an equilateral triangle which means that that all sides are the same length ( same with the one on top ).

So if one side is 9, the other side is also 9.

Thus the answer to your problem is, 9

Answer:

[tex]x = 6 \sqrt{3} [/tex]

The condensed form of the given logarithmic expression ln(3) + ln(x) + ln(y) is: ln(3*x*y)

To condense the given logarithmic expression to a single logarithm with a leading coefficient of 1, we can use the product property of logarithms.

The product property states that the sum of two logarithms with the same base is equivalent to the logarithm of the product of the two numbers.

Using this property, we can combine the three logarithmic terms in the given expression: ln(3) + ln(x) + ln(y) = ln(3*x*y)

Therefore, the condensed form of the given logarithmic expression is:
ln(3*x*y). This is a single logarithm with a leading coefficient of 1, as required.

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Ben's ball lands approximately 2.5 seconds after Andrew's ball.

Since the value of parameter a is -5 for both balls, the height of each ball follows the equation:

h(t) = -5t² + vt + h0

where;

Let's assume that Andrew's ball is hit with an initial velocity of v1, and Ben's ball is hit with an initial velocity of v₂. We also know that Ben's ball reaches a maximum height 50% greater than Andrew's, which means that:

h_max2 = 1.5h_max1

At the maximum height, the velocity of the ball is zero, so we can find the time it takes for each ball to reach the maximum height by setting v = 0 in the equation for h(t):

t_max1 = v₁ / (2 x 5)

t_max2 = v₂ / (2 x 5)

Since Ben's ball reaches a maximum height that is 50% greater than Andrew's, we can write:

h_max2 = 1.5h_max1

-5(t_max2)² + v₂t_max2 + h0 = 1.5(-5(t_max1)² + v1 * t_max1 + h0)

Simplifying this equation, we get:

-5(t_max2)² + v₂t_max2 = -7.5(t_max1)² + 1.5v₁t_max1

We also know that Andrew's ball lands after 4 seconds, which means that h(4) = 0:

h(4) = -5(4)² + v1 * 4 = 0

-80 + 4v1 = 0

v1 = 80/4

v1 = 20 m/s

Solving these equations for t_max2 and v2, we get:

t_max1 = v1 / (2 x 5)

t_max1 = 20 / (2 x 5)  = 2 s

t_max2 = 1.5 * t_max1 = 3 s

v2 = 1.5 * v1 = 30 m/s

To find the time it takes for Ben's ball to land, we need to find the time t2 when h(t2) = 0.

We can use the equation for h(t) with v = v2, h0 = 0, and solve for t:

-5t² + v₂t = 0

-5t² + 30 = 0

5t² = 30

t² = 30/5

t² = 6

t = √6

t = 2.5 s

Therefore, Ben's ball lands approximately 2.5 seconds after Andrew's ball.

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Answer:

The given rectangle is not fully defined, as it is missing some measurements such as the length and width. Without this information, it is not possible to accurately calculate the perimeter of the rectangle.

However, assuming that the missing measurement is the width of the rectangle, then the expressions given by the five students would be:

2(3x + 4x + 3) = 14x + 6

2(6x + 6) + 2(4x + 6) = 20x + 24

2(3x + 3) + 2(4x + 3) = 14x + 12

2(6x + 3) + 2(4x + 3) = 20x + 12

2(6x + 3x) + 2(3 + 4x) = 18x + 10

Note that all expressions follow the formula for the perimeter of a rectangle, which is P = 2l + 2w, where l is the length and w is the width of the rectangle.

Answer:

To find the height of the plane, we need to use trigonometry. Let's call the height of the plane "h". We can use the given angle of 57° and the opposite side (height) to the angle to find the adjacent side (distance traveled east) using the tangent function:

tan(57°) = h / distance traveled east

We can rearrange this equation to solve for h:

h = distance traveled east x tan(57°)

To find the distance traveled east, we need to use the given distance of 322 miles and the direction traveled. Since the plane is traveling at a 57° angle northeast, we can split this into two right triangles, one facing northeast and the other facing southeast, as shown below:

N

|

|

|\ 57°

|

\

The distance traveled east is the adjacent side of the southeast-facing right triangle, which can be found using the cosine function:

cos(57°) = distance traveled east / 322

We can rearrange this equation to solve for the distance traveled east:

distance traveled east = 322 x cos(57°)

Now we can plug in this value for the distance traveled east into the equation for the height of the plane:

h = distance traveled east x tan(57°)

h = (322 x cos(57°)) x tan(57°)

Using a calculator, we can evaluate this expression to find:

h ≈ 389.4 miles

Therefore, the height of the plane to the nearest mile is 389 miles.

The standard matrix for the stated composition of linear operators on R2 is:

The standard matrix for the stated composition of linear operators on R2 can be found by multiplying the matrices for each individual operation.

First, let's find the matrix for a rotation of 270° counterclockwise:

Answer:

A, I and III

Step-by-step explanation:

The origin is the point in the center of a graph, where the x- and y-axes intersect. This point is also (0,0).

V(t)=S ert-W/r

a. The rate at which the account changes (dt/dV) is equal to the interest rate (r) times the value of the account (V) minus the amount withdrawn (W). This can be written mathematically as dt/dV=rV-W. This is a separable differential equation, which can be solved by integrating both sides of the equation with respect to time. The solution is V(t)=S ert-W/r.

b. The value of the Jones' account at the end of 10 years can be found using the solution V(t) from part a: V(10)=500,000 e5%x10-50,000/5% = $387,468.51.

c. To keep their account unchanged at $500,000, the Jones' must withdraw an annual amount W such that V(t)=500,000. From the solution V(t)=S ert-W/r, we can solve for W: W = 500,000e5%x10 - 500,000/5% = $47,613.30.

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Answer:

approximately 10.3 days to reach 500 bacteria

Step-by-step explanation:

Let's use the formula for exponential growth to write an equation that represents the situation: N = N0 * e^(rt)

where N is the number of bacteria, N0 is the initial number of bacteria, e is Euler's number (approximately 2.718), r is the growth rate, and t is the time in days.

We know that after 4 days, the number of bacteria is 71, so we can plug these values into the equation to solve for the growth rate: 71 = N0 * e^(4r)

Similarly, after 7 days (4 + 3), the number of bacteria is 185: 185 = N0 * e^(7r)

Now we have two equations with two unknowns (N0 and r). We can divide the second equation by the first equation to eliminate N0: 185/71 = e^(3r)

Taking the natural logarithm of both sides, we get: ln(185/71) = 3r

Solving for r, we get: r = ln(185/71) / 3 ≈ 0.558

Now we can use the first equation and the growth rate we just found to solve for N0:

71 = N0 * e^(4 * 0.558)

N0 ≈ 11.7

So the initial number of bacteria was approximately 11.7.

To find out how long it will take to reach 500 bacteria, we can plug in the values we know into the equation and solve for t: 500 = 11.7 * e^(0.558t)

Dividing both sides by 11.7, we get: e^(0.558t) ≈ 42.74

Taking the natural logarithm of both sides, we get: 0.558t ≈ ln(42.74)

Solving for t, we get: t ≈ ln(42.74) / 0.558 ≈ 10.3 days

Therefore, it will take approximately 10.3 days for the sandwich to be fully covered with bacteria.

The constant term is 8, which means that the y-intercept of the graph of g is (0, 8).

In mathematics, a function is a rule that assigns a unique output value for every input value in a set. It is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output. Functions are widely used in various fields of mathematics, science, engineering, and technology to model and analyze real-world situations, to describe how quantities depend on one another, and to solve problems.

Here,

If we set k=4, the function g(x) becomes:

g(x) = k(x+2) = 4(x+2)

The value of k affects the slope of the graph of g compared to the graph of f. In this case, since k=4, the slope of the graph of g is 4 times the slope of the graph of f.

The slope of the graph of f is 1, since the coefficient of x is 1. Therefore, the slope of the graph of g is:

4 * 1 = 4

This means that the graph of g is steeper than the graph of f.

The y-intercept of the graph of f is 2, since the constant term is 2. Setting k=4 does not affect the y-intercept of the graph of g, since the constant term remains the same:

g(x) = 4(x+2) = 4x + 8

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The equation of the horizontal asymptote of P isy = 22818.18.  This means that the insect population will approach a maximum value of approximately 22818 insects, no matter how many months have passed since they were introduced. The unit of measure for this value is insects per month.

a. The equation of the horizontal asymptote can be found by taking the limit of P(t) as t approaches infinity.

lim P(t) as t approaches infinity = lim (50+251)/(2+0.011t) as t approaches infinity

Since the denominator grows without bound as t approaches infinity, the limit of P(t) approaches 251/0.011 = 22818.18. Therefore, the equation of the horizontal asymptote is:

y = 22818.18

b. The horizontal asymptote tells us the maximum population that the protected area can sustain for this type of insect. In other words, as time goes on, the insect population will approach a maximum value of approximately 22818 insects. The unit of measure for the asymptote value is insects.

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The expression ((-8x⁵)(x⁻³))/(20x²) when simplified, making all exponents positive, will become -2/5.

To simplify the expression ((-8x⁵)(x⁻³))/(20x²), we need to apply the rules of exponents and simplify the coefficients.

First, let's simplify the coefficients:
-8/20 = -2/5

Next, let's apply the rules of exponents:

Product Rule: When multiplying two exponential expressions with the same base, you can add the exponents. That is, xᵃ · xᵇ = xᵃ⁺ᵇ

Quotient Rule: When dividing two exponential expressions with the same base, you can subtract the exponents. That is, xᵃ / xᵇ = xᵃ⁻ᵇ

(x⁵)(x⁻³) = x⁵⁺⁽⁻³⁾ = x²
x²/x² = x²⁻² = x⁰ = 1

So the expression simplifies to:
(-2/5)(1) = -2/5

Therefore, the simplified expression is -2/5.

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The inequality of the scenario is 8.50n + 9 ≤ 43

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

Amount = $43

The cost of n books and 4 pens can be expressed as:

8.50n + 2.25(4)

We can simplify this expression:

8.50n + 9

We know the student has a total of $43 to spend, so we can set up an inequality:

8.50n + 9 ≤ 43

Subtracting 9 from both sides, we get:

8.50n ≤ 34

Dividing both sides by 8.50, we get:

n ≤ 4

Hence, the inequality that best represents the scenario is: n ≤ 4 or 8.50n + 9 ≤ 43

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Complete question

For this item, select the answers from the drop-down menus A student has $43 in total to buy books that cost \$8.50 each and pens that cost $2.25 each. The student buys n number of books and 4 pens

Complete the inequality to best represent the scenario.

The correct answer is C. the probability of incorrectly rejecting the null hypothesis.

A P-value is used in hypothesis testing to determine the likelihood of observing a test statistic as extreme as the one observed under the null hypothesis. It is a measure of the strength of evidence against the null hypothesis.

A small P-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject the null hypothesis. A large P-value (> 0.05) indicates weak evidence against the null hypothesis, so you fail to reject the null hypothesis.

It is important to note that a P-value is not the same as the significance level (option E), which is the threshold used to determine whether to reject or fail to reject the null hypothesis.

The P-value is also not the same as the correlation between two variables (option A), which measures the strength of a linear relationship between two variables.

The P-value is also not the ratio between the test statistic and the standard error (option B), which is used to calculate the P-value but is not the same thing.

Therefore, the correct answer is option C, the probability of incorrectly rejecting the null hypothesis.

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Instead of regular six-sided dice, some games use dodecahedronal - or 12-sided - dice. If you rolled a pair of dodecahedronal dice (each label 1 through 12) 100 times, you would expect the values on the two dice to add up to 4about 3 times. The correct answer is c) 3.


When rolling two dodecahedronal dice, there are a total of 12 x 12 = 144 possible outcomes. To find the probability of rolling a sum of 4, we need to look at the possible combinations that can result in a sum of 4:

1 + 3

2 + 2

3 + 1

There are a total of 3 possible combinations that can result in a sum of 4. Therefore, the probability of rolling a sum of 4 is 3/144 = 1/48.


If we roll the dice 100 times, we would expect to get a sum of 4 about (1/48) x 100 = 2.08333 times. Since we can't roll a fraction of a time, we can round this to the nearest whole number, which is 3. So, we would expect to roll a sum of 4 about 3 times out of 100 rolls.The correct answer is c) 3.

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