The table shows the median annual salaries for two different jobs. Median Annual Salaries Job Marketing Manager Financial Analyst Median Annual Salary (dollars) 115,750 76,950 Based on the information in the table, how much more money would a marketing manager earn than a financial analyst over 10 years.
F $38,800
G $1,927,000
H $388,000
J $192,700
pls help!​

Answer:

Pick c, 2/27

Step-by-step explanation:

Bag #1 has 4 + 6 + 3 + 5 = 18 total tiles.

4 are black.

So prob (black) from bag 1 = 4/18.

Bag #2 has 3 + 2 + 3 + 1 = 9 total tiles.

3 are black.

So prob(black) from bag 2 = 3/9.

Multiply those 2 independent probabilities together.

(4/18) x (3/9) = (2/27)

It can be said that  the statement that correctly compares the function shown on this graph with the function y =4x + 2 is

'The expresion indicated on the graph has a smaller rate of change, but a higher starting point.' (Option A)

We have the expression - y = mx + c, where m is the slope and

c is known as the y-intercept"

For given question,

We have been given a expresion  which states: y = 4x + 2

This function represents a line.

Looking at the above function side by side with slop-intercept form of the line.

We can state that we have slope of the line = 4 and y - intercept = 2

Note that the pace of change of the line is given by the slope.

This means that  the required pace of change of function y = 4x + 2 will be 4.

A beginning point of a function is the commencement value in a linear expression .

The beginning point of above function f(x) is 2.

The graph of the line passes via points (-1, 1) and (0, 4)

So using slope formula,

m1 = (4-1)/(0-(-1))

m1 = 3

This means that  the pace of change of function shown on the graph is 3.

This means, the pace of change of function shown on the graph is smaller than the pace of change of function y = 4x + 2

The expression of the function shown on the graph would be y = 3x + 4

The point of incetion of the function shown on the graph is 4.

This signifies that the starting point of the graphed function is higher than the starting point of the function y = 4x + 2.

As a result, the accurate statement comparing the function displayed on this graph to the function y=4x + 2 is 'The function shown on the graph has a lesser rate of change, but a higher beginning point.'

The correct answer is option (A)

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To measure the length of the segment, place a ruler measured in centimeters on the surface and record the length of choice.

If you want to determine any line segment, use a ruler that is calibrated in centimeters (as is the case in the above), inches, or meters. Place the ruler on the paper or drawing platform and draw a line that matches the corresponding figure.

If the measure of the line segment is 8.3 cm then trace that length on the surface. To bisect the line segment, draw a vertical line across the horizontal one and divide the opposite sides by 2.

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Answer: 0.71

Step-by-step explanation:

[tex]a^2+b^2=c^2\\a=b\\so\\2a^2=c^2\\c=1 here\\2a^2=1^2\\a=\sqrt{1/2} \\a=0.71[/tex]

The logarithmic expressions when expanded are

1. [tex]\log2^{8yz}[/tex] = 8xyz log(2)

2. [tex]\log9^{81x/y}[/tex] = 81x/y log(9)

3. [tex]\log5^{x^3}[/tex] = x³ log(5)

4. [tex]\log6^{\sqrt[3]{x}}[/tex] = ∛x log(6)

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

The logarithmic expressions

The logarithmic expressions can be expanded using power rule of logarithm which states that

logaᵇ = b log(a)

Using the above as a guide, we have the following:

[tex]\log2^{8yz}[/tex] = 8xyz log(2)

[tex]\log9^{81x/y}[/tex] = 81x/y log(9)

[tex]\log5^{x^3}[/tex] = x³ log(5)

[tex]\log6^{\sqrt[3]{x}}[/tex] = ∛x log(6)

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The location of the angles and the parallel lines [tex]\overline{BE}[/tex] and [tex]\overline{ADE}[/tex] indicates;

(a) ΔABC ~ ΔECA by Angle Angle similarity

(b) The ratio of corresponding sides in similar triangles indicates; BC/CA = AB/EC

Parallel lines are lines that continues indefinitely, maintaining the same distance between each other.

9. The specified dimensions of the geometric figures are;

[tex]\overline{EC}[/tex] is a tangent to the circle O

[tex]\overline{AC}[/tex] is a diameter of the circle

[tex]\overline{BC}[/tex] is parallel to secant [tex]\overline{ADE}[/tex]

Therefore, the angle ∠ABC is a right angle (Angle at the circumference formed by the diameter of a circle

∠ACE = 90° (The tangent is perpendicular to the radius of a circle)

∠ABC ≅ ∠ACE (Definition of congruent angles)

m∠ABC = m∠ACE = 90° (Definition of congruence)

∠BCA ≅ ∠EAC (Alternate interior angles)

(b) The similarity between the triangles and the ratio of the corresponding sides indicates;

BC/CA = AC/AE

Therefore

Segment BC in triangle ΔABC corresponds to segment CA in triangle ΔECA

Segment CA in triangle ΔABC corresponds to segment AE in triangle ΔECA

Which indicates;

Segment AB in triangle ΔABC corresponds to segment EC in triangle ΔECA

The ratio of the corresponding sides is therefore;

BC/CA = AB/EC

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Step-by-step explanation:

Volume of the square pyramid = 1/3 * base area * height

                                                    = 1/3 (3 x 3) * 4 = 12 yd^3

Based on the calculations, the yield on this corporate bond is found as 9.14%

Given as For a $1000 face value purchased at a discount price of $850, if it pays 6% fixed interest for the duration of the bond is the yield on a corporate bond mathematically given as

Yield = 6.5%

Interest paid = value of bond x Interest rate

Interest paid = 1000 * 6%

Interest paid = 60

Therefore

Yield = Interest paid / Price paid

Yield = (60 / 850)x 100

Yield = 9.14%

In conclusion, the yield on a corporate bond is

Yield = 9.14%

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Answer: 7.06

Step-by-step explanation: accelus

Step-by-step explanation:

Domain : (-5,4]

Range [-2,0)

Answer:

OD) x = -2, -5

Step-by-step explanation:

To solve the quadratic equation x^2 + 7x + 10 = 0, we can use factoring or the quadratic formula.

Using factoring:

We need to find two numbers whose sum is 7 and whose product is 10. These numbers are 2 and 5. So, we can write x^2 + 7x + 10 = 0 as (x + 2)(x + 5) = 0.

Setting each factor to 0, we get x + 2 = 0 or x + 5 = 0. Solving for x, we get x = -2 or x = -5.

Using the quadratic formula:

The quadratic formula is x = (-b ± sqrt(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation. In this case, a = 1, b = 7, and c = 10.

Substituting these values into the formula, we get x = (-7 ± sqrt(7^2 - 4(1)(10))) / 2(1)

Simplifying, we get x = (-7 ± sqrt(9)) / 2

Therefore, x = (-7 + 3) / 2 or x = (-7 - 3) / 2

Solving for x, we get x = -2 or x = -5.

Thus, the solutions of the quadratic equation x^2 + 7x + 10 = 0 are x = -2 or x = -5.

Therefore, the answer is (OD) x = -2, -5.

The area of the circle is A = 154 m²

Given data ,

Let the diameter of the circle be d = 14 m

So , the radius of the circle is r = d/2

r = 7 m

Now , area of circle is A = πr²

On simplifying , we get

A = ( 3.14 ) ( 7 )²

A = 154 m²

Hence , the area of circle is A = 154 m²

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The probability that the selection will be P(pink, blue) is: 9/64

The total number of sections on the spinner are 8 sections.

Now, out of the 8 sections, the divisions are as follows:

Yellow sections = 2

Blue sections = 3

Pink sections = 3

Thus:

P(first is pink) = 3/8

P(second is blue) = 3/8

Thus:

P(pink, blue) = (3/8) * (3/8)

= 9/64

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The operations between two functions:

Case 1: f(x) + g(x) = 2 - x - x²

Case 2: g(x) - f(x) = x² - x

Case 3: g(x) · f(x) = (1 - x) · (1 - x²) = 1 - x - x² + x³

Case 4: f(x) - g(x) = x - x²

In this problem we need to perform operations between two functions, one operator for each case. There are three operations used in this problem:

Addition

(f + g) (x) = f(x) + g(x)

Subtraction

(f - g) (x) = f(x) - g(x)

Multiplication

(f · g) (x) = f(x) · g(x)

If we know that f(x) = 1 - x² and g(x) = 1 - x, then the operations between functions are:

Case 1

f(x) + g(x) = 2 - x - x²

Case 2

g(x) - f(x) = x² - x

Case 3

g(x) · f(x) = (1 - x) · (1 - x²) = 1 - x - x² + x³

Case 4
f(x) - g(x) = x - x²

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Answer:

[tex]\bold{V=7048.48 ft^3}[/tex]

Step-by-step explanation:

Solution Given:

radius(r)=12

In right-angled triangle

Hypotenuse = 18

base angle =60°

Note:
In a right-angled triangle, we can use the trigonometric ratios sine, cosine, and tangent to relate the angles and sides of the triangle.
Here are the trigonometric ratios for a right-angled triangle:

Sine (sin): The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

[tex]\boxed{\bold{sin(A) =\frac{ opposite}{hypotenuse}}}[/tex]

Cosine (cos): The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

[tex]\boxed{\bold{cos(A) = \frac{adjacent}{hypotenuse}}}[/tex]

Tangent (tan): The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

[tex]\boxed{\bold{tan(A) =\frac{ opposite}{adjacent}}}[/tex]

Now for the question:

Over here the relation between opposite or height and hypotenuse is given by Sine, So using this formula:

[tex]\bold{sin(60^0) =\frac{ opposite\:or\ height}{18}}[/tex]

doing criss-cross multiplication:

Sin 60°*18=opposite or height

[tex]\frac{\sqrt{3}}{2}*18[/tex] =opposite or height

opposite or height(h)=[tex]9\sqrt{3}[/tex]

Again,

We have

[tex]\boxed{\bold{Volume\:of\:cylinder= \pi * r^2 * h}}[/tex]

where:

π (pi) is a mathematical constant approximately equal to 3.14

r is the radius of the base of the cylinder

h is the height of the cylinder

Now Substituting value

[tex]Volume = \pi * r^2 * h\\Volume=3.14*12^2*9\sqrt{3}\\Volume=\bold{7048.48 ft^3}[/tex]

The equation of the line is y = -4x - 11.

The equation of the line is y = 3x + 7.

The equation of the line is y = (1/2)x + 12.

We have,

A)

The equation of the line with gradient m = -4 that passes through the point (-2,-3) is:

y - y1 = m(x - x1)

y - (-3) = -4(x - (-2))

y + 3 = -4(x + 2)

y + 3 = -4x - 8

y = -4x - 11

B)

The equation of the line with gradient m=3 that passes through the point (-2,1) is:

y - y1 = m(x - x1)

y - 1 = 3(x - (-2))

y - 1 = 3(x + 2)

y = 3x + 7

C)

The equation of the line with gradient m=1/2 that passes through the point (-4,10) is:

y - y1 = m(x - x1)

y - 10 = (1/2)(x - (-4))

y - 10 = (1/2)(x + 4)

y = (1/2)x + 12

Therefore,

The equation of the line is y = -4x - 11.

The equation of the line is y = 3x + 7.

The equation of the line is y = (1/2)x + 12.

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Using iterations, the solution to the above equation  6⁽⁻ˣ⁾ +4 = 3x -1 is Option C  = 27/16. See the graph attached.

The question requires you to state the solution of the equation. On the graph, this would be the point of intersection of both curves.

To solve for x, we'll continue using an iterative method called the fixed-point iteration ..

Rewrite the equation in the form x = g(x):

g(x) = (6⁽⁻ˣ⁾ + 5) / 3

Start with an initial guess, let's say x0 = 1.

Iterate using the formula x(n+1) = g( x(n )) until convergence, where n is the iteration number:

x (1) = g(x 0)

x (2 ) = g (x(1))

x( 3) = g(x (2))

Let's perform three iterations to approximate the solution

Iteration 1

x (1) = g( x0) = (6⁻¹ + 5) / 3

= (1/ 6 +5) / 3

= (1 /6 + 30/6) / 3

= 31/18

≈ 1.7222

Second iteration is

x(2) = g(x (1)) = ([tex]6^{1.72222}[/tex] + 5) / 3 ≈ 1.6806

Iteration 3:

x(3) = g(x (2)) = ([tex]6^-1.6806[/tex] + 5) / 3 ≈ 1.6875

After three iterations, the approximate solution to the equation 6⁽⁻ˣ⁾ + 4 = 3x - 1 is x ≈ 1.6875, which can also be expressed as the fraction 27/16.

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The square root of 107.6 is approximately 10.372

To find the square root of 107.6, we can use a calculator or use the following procedure:

Make an initial estimate:

We know that the square root of 100 is 10, so we can estimate that the square root of 107.6 is slightly more than 10.

Use the formula: We can use the following formula to improve our estimate:

x(n+1) = (x(n) + a/x(n))/2

where x(n) is the nth estimate of the square root of a, and a is the number we want to find the square root of.

Apply the formula: Using the estimate of 10, we get:

x(1) = (10 + 107.6/10)/2 = 10.38

Now we use this new estimate to get a better one:

x(2) = (10.38 + 107.6/10.38)/2 = 10.371

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The publisher must produce and sell approximately 4,857 books to break even.

Let's denote the number of books produced and sold as "x."

The total fixed cost is given as $51,000.

The variable cost per book is $1.50, and since x books are produced and sold, the total variable cost would be 1.50x.

The selling price per book is $12, and since x books are sold, the total revenue would be 12x.

To break even, the total revenue must equal the total cost:

Total Revenue = Total Cost

12x = 51,000 + 1.50x

Subtracting 1.50x from both sides:

12x - 1.50x = 51,000

10.50x = 51,000

Dividing both sides by 10.50:

x = 51,000 / 10.50

x ≈ 4,857

Therefore, the publisher must produce and sell approximately 4,857 books to break even.

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Answer:

b. 5, 11, 6

Step-by-step explanation:

Let the sides of a triangle be a, b, and c, where a < b < c. Then, we know that a + b > c. If we use this identity to solve this problem, then we can find that answer choice b does not satisfy it. Thus, b does NOT  have the lengths of a triangle.

Using the commutative property we can get an equation that depends on the sum of the two variables:

(M + N) = (62 - 2N)/29

The commutative property of the multiplication and sum means that we can permutate the variables:

A*B = B*A

A + B = B + A

In this case, we have the equation:

29*M = 31*(N + 2)

We can distribute the product to get:

29*M = 31*N + 31*2

29*M = 31*N + 62

Now we want to find the value of the sum of the variables, because we have only one equation we can't do that (two variables means that we need two equations) But we can write an expression that depends on the sum of the two variables.

29M + 31N = 62

And now use the commutative property to write:

(29M + 29N) + 2N = 62

(M + N) = (62 - 2N)/29

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The statement that the [tex]x^ky^{n-k}[/tex] in the expansion of ( x + y )ⁿ equals [tex]\binom{k}{n}[/tex] is False.

The coefficient of the term [tex]x^ky^{n-k}[/tex] in the expansion of ( x + y )ⁿ  is given by the binomial coefficient C ( n, k ), not C ( k, n ) . It should therefore be written as C ( n, k ), not C ( k, n ).

Stated alternatively, the quantity in question is the combination of k items selected from a pool of n items, rather than the inverse arrangement. As a result, the accurate binomial coefficient has been determined such that:

C ( n, k ) = n! / [ k! ( n - k ) ! ], where " ! " denotes the factorial function.

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The question is:

The coefficient of [tex]x^ky^{n-k}[/tex] in the expansion of ( x + y )ⁿ equals [tex]\binom{k}{n}[/tex]

True

False.

Answer:

[tex]2-\sqrt{3}[/tex]

Step-by-step explanation:

Find the exact value of the expression, tan(15°).

The method I am about to show you will allow you to solve this problem without any tables or calculators. Although, memorizing the unit circle and trigonometric identities is required.

[tex]\tan(15 \textdegree)\\\\\Longrightarrow \tan(\frac{30 \textdegree}{2} )\\\\\text{Use the half-angle identity:} \ \tan(\frac{A}{2})=\pm \sqrt{\frac{1-\cos(A)}{1+\cos(A)} }=\frac{\sin(A)}{1+\cos(A)} =\frac{1-\cos(A)}{\sin(A)} \\\\\Longrightarrow\frac{1-\cos(30 \textdegree)}{\sin(30 \textdegree)} \\\\\text{From the unit circle:} \ \cos(30 \textdegree)=\frac{\sqrt{3} }{2} \ \text{and} \ \sin(30 \textdegree)=\frac{1}{2}\\[/tex]

[tex]\Longrightarrow \frac{1-\frac{\sqrt{3} }{2}}{\frac{1}{2}}\\\\\Longrightarrow 2(1-\frac{\sqrt{3} }{2})\\\\\therefore \boxed{\boxed{\tan(15 \textdegree)=2-\sqrt{3} }}[/tex]

Thus, the problem is solved.

Answer:

[tex]\tan 15^{\circ} = 2 - \sqrt{3}[/tex]

Step-by-step explanation:

To find the exact value of tan 15°, we can use trigonometric identities and the unit circle.

We know that tan(x) can be expressed as the ratio of sin(x) and cos(x). We can also write 15° as (60° - 45°).

Therefore, tan 15° can be expressed as:

[tex]\tan15^{\circ}=\tan(60^{\circ}-45^{\circ})=\dfrac{\sin(60^{\circ}-45^{\circ})}{\cos(60^{\circ}-45^{\circ})}[/tex]

Now use the trigonometric angle identities to rewrite the ratio in terms of sin 60°, cos 60°, sin 45° and cos 45°.

[tex]\boxed{\begin{minipage}{6.5 cm}\underline{Trigonometric Angle Identities}\\\\$\sin (A - B)=\sin A \cos B - \cos A \sin B$\\\\$\cos (A - B)=\cos A \cos B + \sin A \sin B$\\\end{minipage}}[/tex]

Therefore:

[tex]\begin{aligned}\tan15^{\circ}&=\dfrac{\sin(60^{\circ}-45^{\circ})}{\cos(60^{\circ}-45^{\circ})}\\\\&=\dfrac{\sin60^{\circ}\cos45^{\circ}-\cos60^{\circ}\sin45^{\circ}}{\cos 60^{\circ} \cos 45^{\circ}+ \sin 60^{\circ}\sin 45^{\circ}}\end{aligned}[/tex]

In the unit circle, the cosine of an angle is represented by the x-coordinate of a point on the circle, and the sine of an angle is represented by the y-coordinate of that same point → (x, y) = (cos θ, sin θ). Therefore, we can use the unit circle to identity the values of sin 60°, cos 60°, sin 45° and cos 45°:

[tex]\sin 60^{\circ}=\dfrac{\sqrt{3}}{2}[/tex]

[tex]\cos 60^{\circ}=\dfrac{1}{2}[/tex]

[tex]\sin 45^{\circ}=\dfrac{\sqrt{2}}{2}[/tex]

[tex]\cos 45^{\circ}=\dfrac{\sqrt{2}}{2}[/tex]

Substitute these into the equation and simplify:

[tex]\begin{aligned}\tan15^{\circ}&=\dfrac{\sin(60^{\circ}-45^{\circ})}{\cos(60^{\circ}-45^{\circ})}\\\\&=\dfrac{\sin60^{\circ}\cos45^{\circ}-\cos60^{\circ}\sin45^{\circ}}{\cos 60^{\circ} \cos 45^{\circ}+ \sin 60^{\circ}\sin 45^{\circ}}\\\\&=\dfrac{\dfrac{\sqrt{3}}{2}\cdot \dfrac{\sqrt{2}}{2}-\dfrac{1}{2}\cdot \dfrac{\sqrt{2}}{2}}{\dfrac{1}{2}\cdot \dfrac{\sqrt{2}}{2}+ \dfrac{\sqrt{3}}{2}\cdot \dfrac{\sqrt{2}}{2}}\\\\\end{aligned}[/tex]

           [tex]\begin{aligned}&=\dfrac{\dfrac{\sqrt{2}}{2} \left(\dfrac{\sqrt{3}}{2}-\dfrac{1}{2}\right)}{\dfrac{\sqrt{2}}{2} \left(\dfrac{1}{2}+ \dfrac{\sqrt{3}}{2}\right)}\\\\&=\dfrac{\dfrac{\sqrt{3}}{2}-\dfrac{1}{2}}{ \dfrac{1}{2}+ \dfrac{\sqrt{3}}{2}}\\\\&=\dfrac{\dfrac{\sqrt{3}-1}{2}}{\dfrac{1+\sqrt{3}}{2}}\\\\&=\dfrac{\sqrt{3}-1}{1+\sqrt{3}}\end{aligned}[/tex]

Simplify further by multiplying the numerator and denominator by the conjugate of the denominator:

           [tex]\begin{aligned}&=\dfrac{\sqrt{3}-1}{1+\sqrt{3}}\cdot \dfrac{1-\sqrt{3}}{1-\sqrt{3}}\\\\&=\dfrac{(\sqrt{3}-1)(1-\sqrt{3})}{(1+\sqrt{3})(1-\sqrt{3})}\\\\&=\dfrac{\sqrt{3}-3-1+\sqrt{3}}{1-\sqrt{3}+\sqrt{3}-3}\\\\&=\dfrac{2\sqrt{3}-4}{-2}\\\\&=-\sqrt{3}+2\\\\&=2-\sqrt{3}\end{aligned}[/tex]

Therefore, the exact value of tan 15° is (2 - √3).

An inequality to describe the situation is 3x + 2y ≤ 30.

We are given that;

Cost of hamburger meat= $3

Cost of hot dog=$2

Now,

The total cost of hamburger meat and hot dogs is no more than $30. The variable x represents the number of pounds of hamburger meat, and the variable y represents the number of pounds of hot dogs. The coefficient 3 is the cost per pound of hamburger meat, and the coefficient 2 is the cost per pound of hot dogs.

Therefore, by the inequality the answer will be 3x + 2y ≤ 30

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The average cost function for the given total cost function is AC(x) = 2,100/x + 4 - 0.0003x.

To find the average cost function C, we need to divide the total cost function C(x) by the quantity of output produced, which is represented by x.
The formula for average cost (AC) is:
AC(x) = C(x) / x
Plugging in the given values, we have:
AC(x) = (2,100 + 4x - 0.0003x2) / x
Simplifying this expression, we get:
AC(x) = 2,100/x + 4 - 0.0003x
Therefore, the average cost function C(x) is:
C(x) = 2,100/x + 4x - 0.0003x2
This function represents the average cost per unit of output, taking into account fixed costs (2,100) and variable costs (4x - 0.0003x) that increase as output increases.
It's important to note that the cost function C(x) is quadratic, which means that the average cost function C(x) will have a U-shaped curve.

This is because initially, as output increases, fixed costs are spread out over a larger quantity of output, leading to a decrease in average cost.

However, at a certain point, the increasing variable costs will outweigh the decreasing fixed costs, causing average cost to increase again.
Overall, knowing the average cost function can be useful for businesses to make decisions about pricing, production levels, and cost management strategies.

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a) there are 8,535,316 ways to choose a dance committee of 8 students. b) there are 497,070 ways to choose a dance committee consisting of 2 freshmen, 2 sophomores, 2 juniors, and 2 seniors. c) there are 4,410 ways to choose a dance committee consisting of 4 juniors and 4 seniors.

a) To determine the number of ways a dance committee of 8 students can be chosen, we need to consider the total number of eligible students and choose a group of 8 from them.

Total number of eligible students = 7 freshmen + 10 sophomores + 9 juniors + 7 seniors = 33 students

The number of ways to choose a committee of 8 students from a pool of 33 is given by the combination formula:

C(33, 8) = 33! / (8!(33-8)!) = 8,535,316

Therefore, there are 8,535,316 ways to choose a dance committee of 8 students.

b) To choose a dance committee consisting of 2 freshmen, 2 sophomores, 2 juniors, and 2 seniors,

Number of ways to choose 2 freshmen from 7 freshmen = C(7, 2) = 21

Number of ways to choose 2 sophomores from 10 sophomores = C(10, 2) = 45

Number of ways to choose 2 juniors from 9 juniors = C(9, 2) = 36

Number of ways to choose 2 seniors from 7 seniors = C(7, 2) = 21

Total number of ways to choose the dance committee = 21 * 45 * 36 * 21 = 497,070

Therefore, there are 497,070 ways to choose a dance committee consisting of 2 freshmen, 2 sophomores, 2 juniors, and 2 seniors.

c) To choose a dance committee consisting of 4 juniors and 4 seniors,

Number of ways to choose 4 juniors from 9 juniors = C(9, 4) = 126

Number of ways to choose 4 seniors from 7 seniors = C(7, 4) = 35

Total number of ways to choose the dance committee = 126 * 35 = 4,410

Therefore, there are 4,410 ways to choose a dance committee consisting of 4 juniors and 4 seniors.

d) The probability of selecting a committee consisting of 2 freshmen, 2 sophomores, 2 juniors, and 2 seniors

Favorable outcomes = 497,070 (from part b)

Total possible outcomes = 8,535,316 (from part a)

Probability = Favorable outcomes / Total possible outcomes

= 497,070 / 8,535,316

≈ 0.058 (rounded to the nearest thousandth)

Therefore, the probability of selecting a committee consisting of 2 freshmen, 2 sophomores, 2 juniors, and 2 seniors is approximately 0.058.

e) The probability of selecting a committee consisting of 4 juniors and 4 seniors

Favorable outcomes = 4,410 (from part c)

Total possible outcomes = 8,535,316

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The probability that more than two out of the twelve automobiles fail the test is approximately 29. 35%.

The probability of more than two automobiles failing the emissions test would be:

= 1 - Probability that none fails - Probability that 1 fails - Probability that 2 fail

The probabilities are :

P (0 fail) = C (12, 0) x ( 0.08 ⁰) x (0.92 ¹² )

= 1 x 1 x 0. 2785

= 0. 2785

P(1 fail) = C (12, 1) x (0.08 ¹ ) x ( 0.92 ¹¹ )

= 12 x 0. 08 x 0. 3016

= 0. 2899

P (2 fail) = C (12, 2) x (0.08 ² ) x ( 0.92 ¹⁰)

= 66 x 0. 0064 x 0.3259

= 0.1381

The probability that more than two fail is:

= 1 - (0.2785 + 0.2899 + 0.1381)

= 0.2935

= 29. 35 %

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