Let f(x)= -2x+4 and g(x)= 3x^2. Find (f+g)(x) and (f-g)(x)
State the domain of each.
Evaluate the following: (f+g)(-3) and (f-g)(-3)

If P (A) = 0. 5, P(B) = 0. 1, and A and B are mutually exclusive, then P(A or B) is 0.6 or 60%.

To find the probability of A or B occurring, we use the formula P(A or B) = P(A) + P(B) - P(A and B). However, since A and B are mutually exclusive events, they cannot occur together. This means that the probability of A and B occurring together is zero. Therefore, we can simplify the formula to P(A or B) = P(A) + P(B).

Using the given values, we have P(A) = 0.5 and P(B) = 0.1. Plugging these values into the formula, we get:

P(A or B) = 0.5 + 0.1
P(A or B) = 0.6

Therefore, the probability of A or B occurring is 0.6 or 60%. This means that there is a 60% chance of either A or B happening, but not both at the same time since they are mutually exclusive.

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To find the moment of inertia about the y-axis of the first-quadrant area bounded by the curve y=9−x^2 and the coordinate axes, we can use the formula:

I = ∫y² dA

where I is the moment of inertia, y is the distance from the y-axis to the infinitesimal element of area dA, and the integral is taken over the first-quadrant area.

To set up the integral, we need to express y in terms of x for the curve y=9−x². Solving for y, we get:

y = 9 - x²

The area element dA is given by:

dA = y dx

Substituting y in terms of x, we get:

dA = (9 - x²) dx

Now we can express the moment of inertia as an integral:

I = ∫y² dA
 = ∫(9 - x²)² dx     (limits of integration: x = 0 to x = 3)

To evaluate the integral, we can expand the integrand using the binomial theorem:

I = ∫(81 - 36x² + x⁴) dx
 = 81x - 12x³ + (1/5)x⁵     (limits of integration: x = 0 to x = 3)

Finally, we can substitute the limits of integration and simplify:

I = (81(3) - 12(3)³ + (1/5)(3)⁵) - 0
 = 243 - 108 + 27
 = 162

Therefore, the moment of inertia about the y-axis is 162 units^4.
To find the moment of inertia (Iy) about the y-axis for the first-quadrant area bounded by the curve y = 9 - x^2 and the coordinate axes, we need to integrate the expression for the moment of inertia using the limits of the region.

The curve intersects the x-axis when y = 0, so:

0 = 9 - x²
x² = 9
x = ±3

Since we're in the first quadrant, we're interested in x = 3.

The moment of inertia about the y-axis is given by the expression Iy = ∫x²dA, where dA is the area element. In this case, we'll use a vertical strip with thickness dx and height y = 9 - x². Therefore, dA = y dx.

Now, let's integrate Iy:

Iy = ∫x²(9 - x²) dx from 0 to 3

To solve this integral, you may need to use polynomial expansion and integration techniques:

Iy = ∫(9x² - x⁴) dx from 0 to 3
Iy = [3x³/3 - x⁵/5] from 0 to 3
Iy = (3(3)³/3 - (3)⁵/5) - (0)
Iy = (81 - 243/5)
Iy = (405 - 243)/5
Iy = 162/5

So the moment of inertia about the y-axis for the first-quadrant area bounded by the curve y = 9 - x^2 and the coordinate axes is Iy = 162/5.

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We know that the westbound car has traveled 24 miles.

When the northbound car has traveled 18 miles and the straight-line distance between the two cars is 30 miles, you can use the Pythagorean theorem to determine the distance the westbound car has traveled. The theorem states that a² + b² = c², where a and b are the legs of a right triangle and c is the hypotenuse.

In this case, the northbound car's distance (18 miles) represents one leg (a) and the westbound car's distance represents the other leg (b). The straight-line distance between the cars (30 miles) represents the hypotenuse (c). The equation can be set up as follows:

18² + b² = 30²

Solving for b:

324 + b² = 900
b² = 900 - 324
b² = 576
b = √576
b = 24

So, the westbound car has traveled 24 miles.

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To model the volume of a rectangular prism with length 26cm and width w and height h such that the sum of the width and height is 32cm, we can use the following function:

V(w, h) = 26wh

subject to the constraint:

w + h = 32

We can solve for one of the variables in the constraint equation and substitute it into the volume equation, giving us:

w + h = 32  =>  h = 32 - w

V(w) = 26w(32 - w) = 832w - 26w^2

To find the maximum possible volume, we can take the derivative of this function with respect to w and set it equal to zero

dv/dw= 832 - 52w = 0

Solving for w, we get:

w = 16

Substituting this value back into the constraint equation, we get:

h = 32 - w = 16

Therefore, the maximum possible volume of the prism is:

V(16, 16) = 26(16)(16) = 6656 cubic cm

So the function to model the volume of the rectangular prism is V(w) = 832w - 26w^2, and the maximum possible volume is 6656 cubic cm when the width and height are both 16cm.

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The arc angle indicated with ? is derived as 230° using the angle between intersecting tangents.

The angle between two tangent lines which intersect at a point is 180 degrees minus the measure of the arc between the two points of tangency.

angle G = 180° - arc angle HF

arc angle HF = 180° - 50°

arc angle HF = 130°

so the arc angle indicated with ? is;

? = 360° - 130°

? = 230°

Therefore, using the angle between the intersecting tangents, the arc angle indicated with ? is 230°.

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Determine whether the two figures are similar: D. No, the two figures are not similar.

In Geometry, two (2) quadrilaterals are similar when the ratio of their corresponding sides are equal in magnitude and their corresponding angles are congruent.

Additionally, two (2) geometric figures such as quadrilaterals are considered to be congruent only when their corresponding side lengths are congruent (proportional) and the magnitude of their angles are congruent;

Ratio = 12/8 = 10/6 = 10/6

Ratio = 3/2 ≠ 5/3 = 5/3

In conclusion, the two figures are not similar because the ratio of their corresponding sides is not proportional.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Answer:

D) 205

Step-by-step explanation:

If the technician has a total of 2.87 oz, and can have a max of 0.014 oz in each capsule, we have to divide the total amount by the max amount per bottle.

2.87/0.014

=205

This means that the technician can fill 205 capsules with 0.014 oz of vitamin C.

Hope this helps!

The value of tan(ф) when cos(ф) = 8/9 in quadrant IV is - √17/8

To solve for the value of tan(ф), we need to use the trigonometric identity: tan(ф) = sin(ф)/cos(ф).

Since ф is in quadrant IV, we know that the cosine value is positive (due to cosine being positive in the adjacent side of quadrant IV) and the sine value is negative (due to sine being negative in the opposite side of quadrant IV).

We are given the value of the cosine, which is cos(ф) = 8/9. To find the sine, we can use the Pythagorean identity: sin²(ф) + cos²(ф) = 1.

Plugging in the given value of the cosine, we get: sin²(ф) + (8/9)² = 1. Solving for sin(ф), we get sin(ф) = - √(1 - (64/81)) = - √(17/81) = - √17/9.

Now that we have the values of sin(ф) and cos(ф), we can substitute them into the tan(ф) equation: tan(ф) = sin(ф)/cos(ф) = (- √17/9)/(8/9) = - √17/8.

Therefore, the value of tan(ф) when cos(ф) = 8/9 in quadrant IV is - √17/8.

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Lower Bound: 20 - 29.67 ≈ -9.67

Upper Bound: 20 + 29.67 ≈ 49.67.

The conversion of MAD to the standard deviation for normal distributions can be obtained using the given relationship:

Standard Deviation (σ) = MAD / 0.6745

For both distributions, the MAD holds a value of $20, thus arriving at σ ≈ 29.67.

There are two normal distributions now defined by their parameters as follows:

Mean (µ1) = $100 and Standard Deviation (σ1) = 29.67

Mean (µ2) = $120 and Standard Deviation (σ2) = 29.67.

Since both distributions share an equivalent standard deviation, we can perform a comparison of means to determine the overlap between them.

4

Typically there is observed moderate overlapping within one standard deviation from the difference in means.

The calculation of the difference in means indicates µ2 - µ1 = 120 - 100 = 20. Taking one standard deviation (which equates to 29.67) into consideration with respect to the difference of the means leads us to this range:

Lower Bound: 20 - 29.67 ≈ -9.67

Upper Bound: 20 + 29.67 ≈ 49.67.

It's noteworthy that negative values would not make sense within this context leading us to assume that the approximate overlap range is situated between $0 and $50 resulting in these normal distributions manifesting a sensible amount of overlap therein.

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The Complete Question

When comparing two normal distribution graphs with a mean of $100 and $120 respectively, and both having a MAD of $20, at what range do you see a moderate amount of overlap between the two distributions?

The value of the record high temperature is 114°F

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

This means that

High - Low = 109

Substitute the known values in the above equation, so, we have the following representation

High - 5 = 109

Add 5 to both sides

High = 114

Hence, the record high temperature is 114°F

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You're partially correct, as if x approaches ∞ it would approach 2, as eˣ is exponentially growing if x is positive.

If x is negative, which it is in this case, eˣ would get exponentially smaller. For example, e⁻² = 1/e².

So, in this case [tex]\frac{5}{e^x}[/tex] would get exponentially larger, as it is a number over an increasingly small number, like how [tex]\frac{1}{0.001}[/tex] is larger than [tex]\frac{1}{0.1}[/tex].

Therefore the limit would be equivalent to [tex]\frac{2}{\infty}[/tex], which is equal to 0

[tex] \Large{\boxed{\sf \lim_{x\to-\infty}\left(\dfrac{2}{1-\frac{5}{e^x}}\right) = 0}} [/tex]

[tex] \\ [/tex]

Explanation:

We are trying the find the limit of [tex] \: \sf \dfrac{2}{1 - \dfrac{5}{ {e}^{x} } } \: [/tex] when x tends to -∞.

[tex] \\ [/tex]

Given expression:

[tex] \sf \lim_{x\to-\infty}\left(\dfrac{2}{1-\frac{5}{e^x}}\right) [/tex]

[tex] \\ [/tex]

[tex]\blue{\begin{gathered}\begin{gathered} \\ \boxed { \begin{array}{c c} \\ \blue{ \star \: \sf{\boxed{ \sf Properties\text{:}}}} \\ \\ \sf{ \diamond \: \dfrac{c}{ + \infty} = 0^{ + } \: \: and \: \: \dfrac{c}{ - \infty} = 0^{ - } \: \: , \: where \: c \: is \: a \: positive \: number.} \\ \\ \\ \diamond \: \sf \dfrac{c}{ {0}^{ + } } = + \infty \: \: and \: \: \dfrac{c}{ {0}^{ - } } = - \infty \: \: , \: where \: c \: is \: a \: positive \: number.\\ \\ \\ \diamond \: \sf c - \infty = -\infty \: \: and \: \: c + \infty = \infty \: \: ,\: where \: c \: is \: a \: positive \: number. \\ \\ \\ \sf{ \diamond \: \green{e ^{ - \infty} = 0^{+} \: \: and \: \: e ^{ + \infty} = + \infty} } \\ \end{array}}\\\end{gathered} \end{gathered}}[/tex]

[tex] \\ [/tex]

[tex] \\ [/tex]

[tex] \sf \lim_{x\to-\infty}\left(\dfrac{2}{1-\frac{5}{e^x}}\right) = \sf \left(\dfrac{2}{1-\frac{5}{e^{ - \infty}}}\right) [/tex]

[tex] \\ [/tex]

Simplify knowing that [tex] \sf e^{-\infty} \\ [/tex] approaches 0 but remains a positive number. This will be written as 0⁺.

[tex] \sf \lim_{x\to-\infty}\left(\dfrac{2}{1-\frac{5}{e^x}}\right) = \left(\dfrac{2}{1-\frac{5}{e^{ - \infty}}}\right) = \left(\dfrac{2}{1-\frac{5 \: \: }{0^{ + } }}\right)[/tex]

[tex] \\ [/tex]

Simplify again knowing that 5/0⁺ = +∞.

[tex] \\ [/tex]

[tex] \sf \lim_{x\to-\infty}\left(\dfrac{2}{1-\frac{5}{e^x}}\right) = \left(\dfrac{2}{1-\frac{5 \: \: }{0^{ + } }}\right) = \sf \left(\dfrac{2}{1 - \infty}\right) = \dfrac{2}{ - \infty} [/tex]

[tex] \\ [/tex]

[tex] \\ [/tex]

[tex] \sf \lim_{x\to-\infty}\left(\dfrac{2}{1-\frac{5}{e^x}}\right) = \dfrac{2}{ - \infty} = 0^{-} \\ \\ \\ \implies \boxed{ \boxed{ \sf \lim_{x\to-\infty}\left(\dfrac{2}{1-\frac{5}{e^x}}\right) =0}}[/tex]

[tex] \\ \\ \\ [/tex]

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Answer:

If they were both charged today, it would be on the sixth day that they were actually charging at the same time

Answer:

point form - (2,2)

x=2 y=2

Line x is perpendicular to line y. Line z crosses lines x and y. Only statements 3 and 4 are true.

∠6 = ∠8 is not true because they both lie on the same plane and makes an angle of 180° and can never be true. ∠6 = ∠1 is also not true because ∠1 is clearly obtuse angle and ∠6 is clearly acute angle so they cannot be equal. Hence, statement a and b are false.

∠7 = ∠3 is always true because they are corresponding angles and corresponding angles are always equal. m∠2 + m∠4 = 180° is also true because they lie on same plane and have common vertex and hence, they are supplementary angles and make a sum of 180°.  Hence, statement 3 and 4 is always true.

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The function notation is E(h) = 8.5h where h represents the number of hours worked so the domain is {0, 1, 2, 3, 4, 5} and the range is {0, 8.5, 17, 25.5, 34, 42.5}.

Let E(t) be John's earnings in dollars after working t hours, where t is in the domain 0 ≤ t ≤ 5. Then E(t) = 8.50t, since John earns $8.50 per hour proofreading ads.

The domain of the function is 0 ≤ t ≤ 5, since John works no more than 5 hours per day.

The range of the function is 0 ≤ E(t) ≤ 42.50 since John earns $8.50 per hour and works no more than 5 hours per day.

Therefore, the maximum earnings he can make in one day is 5 hours multiplied by $8.50 per hour, which equals $42.50.

The minimum earnings are $0, which would occur if John does not work at all.

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The question is -

John can earn $8.50 per hour proofreading adverse at a local newspaper. He works no more than 5 hours a day. Write a function in function notation and find a reasonable domain and range of his earnings.

The length of the fourth swing is 25.6 inches. The sequence is arithmetic or geometric.

The length of the arc through which a pendulum swings is 50 inches. To determine whether the sequence is arithmetic or geometric, and to find the length of the fourth swing, we will analyze the given information.

The length of the first swing is 50 inches. Each successive swing is 80% of the preceding swing. This means that to find the length of the next swing, we multiply the length of the current swing by 80% (or 0.8).

Since we are multiplying by a constant factor (0.8) to find the next term in the sequence, this is a geometric sequence, not an arithmetic sequence.

Now, let's find the length of the fourth swing.

1st swing: 50 inches
2nd swing: 50 * 0.8 = 40 inches
3rd swing: 40 * 0.8 = 32 inches
4th swing: 32 * 0.8 = 25.6 inches

The length of the fourth swing is 25.6 inches.

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A 95% confidence interval for the true population proportion of people who received flu vaccinations this year is  0.067 < p < 0.133.

To construct a 95% confidence interval for the true population proportion of people who received flu vaccinations, we can use the formula:

CI = p ± z√((p(1-p))/n)

where:

CI is the confidence interval

p is the sample proportion (33/300 = 0.11)

z is the z-score associated with a 95% confidence level, which is approximately 1.96

n is the sample size (300)

Substituting the values, we get:

CI = 0.11 ± 1.96√((0.11(1-0.11))/300)

CI = 0.11 ± 0.043

CI = (0.067, 0.133)

Therefore, the 95% confidence interval for the true population proportion of people who received flu vaccinations is 0.067 < p < 0.133.

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The indicated real nth root(s) of a, where n=3 and a=27 is 3.


You need to find the indicated real nth root(s) of a, where n=3 and a=27. In other words, you need to find the real number(s) that, when raised to the power of 3, equal 27.

Here's a step-by-step explanation:

1. Identify the given values: n=3 and a=27.
2. Write the equation: x^n = a, where x is the real nth root you're trying to find.
3. Substitute the given values: x^3 = 27.
4. Solve the equation for x: x = 3, since 3^3 = 27.

Your answer is x = 3, which is the real 3rd root of 27.

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The area of a rectangle with base of 3x+1 in and a height of 2x-3 in is given as follows:

D. A = 6x² - 7x - 3 in².

The area of a rectangle of length l and width w is given by the multiplication of dimensions, as follows:

A = lw.

The dimensions for this problem are given as follows:

Hence the expression for the area of the rectangle is given as follows:

A = (3x + 1)(2x - 3)

A = 6x² - 9x + 2x - 3

A = 6x² - 7x - 3 in².

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The consumer group's claim that the mean of all possible differences in sample proportions (Brand R minus Brand S) is 0.3 is correct.

This can be calculated by subtracting the sample proportion of Brand S from the sample proportion of Brand R, resulting in a difference of 0.03 or 3%. This matches the consumer group's claim that the mean of all possible differences in sample proportions is 0.3. It is important to note that this result only applies to the specific samples taken and cannot be generalized to all samples from these brands.

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The expression without the summation notation for ∑ 8i+7/n²2 using the summation formulas is (4n + 3)/2n.

To rewrite the expression without the summation notation, we need to use the summation formulas. We can start by expanding the given summation:

∑ 8i+7/n²2 = 8(1)/n²2 + 8(2)/n²2 + 8(3)/n²2 + ... + 8(n)/n²2 + 7/n²2

Next, we can simplify each term by factoring out 8/n²2:

= (8/n²2)(1 + 2 + 3 + ... + n) + 7/n²2

Using the formula for the sum of the first n positive integers, we have:

= (8/n²2)(n(n+1)/2) + 7/n²2

= (4n² + 4n)/2n² + 7/n²2

= (4n + 3)/2n

Therefore, the expression is (4n + 3)/2n.

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The coordinates of point j' after the rotation are (-y, x), where x and y are the original coordinates of point j

The coordinates of point j' after rotating point jkl 90° clockwise around the origin can be found by applying the following transformation:

j' = (cos(90°) * x - sin(90°) * y, sin(90°) * x + cos(90°) * y)

Since we are rotating 90° clockwise around the origin, we have cos(90°) = 0 and sin(90°) = 1, so the transformation simplifies to:

j' = (0 * x - 1 * y, 1 * x + 0 * y)

j' = (-y, x)

Therefore, the coordinates of point j' after the rotation are (-y, x), where x and y are the original coordinates of point j.

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In the given function the domain is [-1, ∞]

Range is [-3, ∞]

The interval when function is positive [0,  ∞]

The domain of a function is the set of values that we are allowed to plug into our function.

This set is the x values in a function such as f(x).

The range of a function is the set of values that the function assumes

In the given function the domain is [-1, ∞]

Range is [-3, ∞]

The interval when function is positive [0,  ∞]

The interval when function is negative [-∞, -1]

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To evaluate the definite integral of the given function, ∫ (t^5 - 2t^2)/t^4 dt, follow these steps:
1. Simplify the integrand: Divide each term by t^4.
  (t^5/t^4) - (2t^2/t^4) = t - 2t^(-2)

2. Integrate each term with respect to t.
  ∫(t dt) - ∫(2t^(-2) dt) = (1/2)t^2 + 2∫(t^(-2) dt)

3. Apply the power rule to the remaining integral.
  (1/2)t^2 + 2(∫t^(-2+1) dt) = (1/2)t^2 + 2(∫t^(-1) dt)

4. Integrate t^(-1) with respect to t.
  (1/2)t^2 + 2(ln|t|)

Now, since we need to evaluate the definite integral, we should have the limits of integration. Let's assume the limits of integration are a and b. Then, apply the Fundamental Theorem of Calculus:

[(1/2)b^2 + 2(ln|b|)] - [(1/2)a^2 + 2(ln|a|)]

This expression gives the value of the definite integral for the given function within the limits a and b.

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The area of the baby blanket with side length of 30 inches is equal to 900 square inches.

Let 'A' represents the area of the square.

And s represents the side length of the square.

The area of a square is given by the formula

A = s^2.

For Meg's baby blanket,

The side length of the baby blanket is equal to 30 inches,

Substitute the values in the area formula we get,

A = s^2

⇒ A = 30^2

⇒ A = 900 square inches

Therefore, the area of Meg's baby blanket is equal to 900 square inches.

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Answer:

Step-by-step explanation:a) To find the price of the car before VAT, we need to first calculate the percentage of VAT included in the price:

VAT% = (VAT / Total Price) x 100

where VAT% is the percentage of VAT, VAT is the value of VAT, and Total Price is the price of the car including VAT.

From the given information, we have:

Total Price = Birr 5,000,000

VAT% = 15% (assuming a VAT rate of 15% in Ethiopia)

Therefore, we can solve for the value of the car before VAT as follows:

Total Price = Car Price + VAT

Birr 5,000,000 = Car Price + 0.15Car Price

Birr 5,000,000 = 1.15Car Price

Car Price = Birr 4,347,826.09

So the price of the car before VAT is Birr 4,347,826.09.

b) To find the value of VAT, we can use the same formula as above and solve for VAT:

Total Price = Car Price + VAT

Birr 5,000,000 = Birr 4,347,826.09 + VAT

VAT = Birr 652,173.91

Therefore, the value of VAT is Birr 652,173.91.

Mollie and Ted can use the Side-Side-Side (SSS) postulate to justify why their triangles must be congruent.

According to the given information, the two triangles share three corresponding sides of equal length: ML = TD, OL = ED, and L = D.

The SSS postulate states that if three corresponding sides of two triangles are congruent, then the triangles are congruent. Therefore, because Mollie's triangle and Ted's triangle share three corresponding sides of equal length, they are congruent by the SSS postulate.

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The calculated value of the expression that is equivalent to √(x¹²qr²) is x⁶r√q

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

√x12qr²

Express properly

So, we have

√(x¹²qr²)

Evaluating the expression in the brackets using the law of indices

So, we have

√(x¹²qr²) = x⁶r√(q)

Next, we open the brackets

This gives

√(x¹²qr²) = x⁶r√q

Hence, the expression that is equivalent to √(x¹²qr²) is x⁶r√q

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To make it to the second round, a participant needs to score approximately 92 (nearest whole number).

To determine the z-value that corresponds to the top 5%, we use the standard normal distribution table. Since we want to find the top 5%, we subtract 5% from 100%, which gives us 95%. The area under the standard normal distribution curve for z-values corresponding to 95% is 1.645 (from the table).

We can use the formula z = (x - μ) / σ to find the score (x) that corresponds to a z-value of 1.645. Plugging in the given values, we get:

1.645 = (x - 76.2) / 17.1

Solving for x, we get x ≈ 91.8. Since we need the score to the nearest whole number, we round up to 92. Therefore, a participant needs to score approximately 92 to make it to the second round.

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There is about a 47.62% or 0.476 chance that the rock hit the windshield in the driver's line of sight. Option D.

To determine the probability that the rock hit the driver's line of sight on the windshield, we need to compare the area of the driver's line of sight rectangle to the total area of the windshield rectangle.

The area of the windshield rectangle is:

A1 = 28 in x 54 in = 1512 sq in

The area of the driver's line of sight rectangle is:

A2 = 30 in x 24 in = 720 sq in

Therefore, the probability that the rock hit the driver's line of sight on the windshield is:

[tex]P= \frac{A2}{A1}= \frac{720 \:sq in}{1512 \:sq in }[/tex] = 0.476 or 47.6%

So, there is about a 47.62% chance that the rock hit the windshield in the driver's line of sight.

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