The total distance in d,in meters, traveled by an object moving in a straight line can be modeled by a quadratic function that is defined in terms of t, is the time in seconds. At a time of 10. 0 seconds the total distance is traveled by the objects is 50. 0 meters and at a time of 20. 0 seconds the total distance traveled by the object is 200. 0 meters if the object was at a distance of 0 meters when t=0 then what is the total distance traveled in meters, by the object after 30. 0 seconds

The maximum population density evaluated is 1200 citizens per square mile, under the condition that the city is 30 miles long and two-thirds as wide, and 555,000 citizens currently live there.

Now to evaluate the maximum population density that is considered in the mayor's calculation is
Let us first calculate the area of the city which is (2/3) × (30 miles)
= 20 miles.
So, now we can calculate the current population density which is
555,000 / (20 × 20)
= 1387.5 citizens per square mile.
Hence the mayor evaluates that at least 75,000 people must transfer out of the city for adequate services to be exercised, we can find the new population as
555,000 - 75,000
= 480,000 citizens.
Therefore, the new population density would be 480,000 / (20 × 20)
= 1200 citizens per square mile

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Option A) Cannot tell with the given information as the question doesn't provide any information about the relationship between having a dog and having a cat.

Without additional information, we cannot determine if these events are independent, dependent, disjoint, or have any other relationship. Independent events are events in which the occurrence or non-occurrence of one event does not affect the occurrence or non-occurrence of the other event.

In other words, the probability of one event happening does not depend on whether or not the other event happens.

Formally, events A and B are independent if and only if:

P(A ∩ B) = P(A) * P(B)

Where P(A) is the probability of event A occurring, P(B) is the probability of event B occurring, and P(A ∩ B) is the probability of both events A and B occurring simultaneously.

If the above equation holds true, then we can say that events A and B are independent. If not, then events A and B are dependent.

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The type of statistical study in which the population is influenced by researchers is known as an experimental study.

In an experimental study, researchers manipulate one or more variables to observe the effect on another variable. The population in an experimental study is usually a sample that is randomly selected to represent the larger population.

The researchers intentionally intervene in the study, which can impact the behavior or responses of the participants. This can be seen as a form of bias since the researchers are influencing the population. However, in some cases, this is necessary to determine causality or to test a hypothesis.

To minimize bias, experimental studies often use control groups. The control group is used to provide a baseline for comparison with the group that is exposed to the manipulated variable. This helps to determine if any observed effects are due to the intervention or if they are due to other factors.

In summary, an experimental study is the type of statistical study in which the population is influenced by researchers. While this can introduce bias, the use of control groups and other measures can help to minimize the impact of this bias on the results.

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The estimate is 7/15.

The given expression is

4/5-1/3

We see that the denominators of both functions are different

So, the numerators can't be added/subtracted directly.

For this, we need to find the equivalent fraction of the given fractions, and the equivalent fractions should have the same denominator.

Now, the denominators are 5 and 3.

To have a common denominator in both fractions, we find the LCM of the denominators.

∴ The LCM of 5 and 3 = 15

Converting the fraction 4/5 into a fraction with 15 as the denominator,

4/5=4×3/5×3=12/15.

The same for 1/3

1/3= 1×5/3×5=5/15

Replacing 4/5 and 1/3 with the equivalent fractions in the given expression, we get,

12/15-5/15=(12-5)/15=7/15

Hence, the estimate is 7/15.

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The average rate of change for the number of shares from 2 minutes to 4 minutes is 25 shares per minute.

To find the average rate of change for the number of shares from 2 minutes to 4 minutes, we need to know the initial number of shares at 2 minutes and the final number of shares at 4 minutes. Once we have those values, we can use the formula:

average rate of change = (final value - initial value) / (time elapsed)

Let's say the initial number of shares at 2 minutes was 100 and the final number of shares at 4 minutes was 150. The time elapsed between 2 minutes and 4 minutes is 2 minutes. Plugging these values into the formula, we get:

average rate of change = (150 - 100) / 2
average rate of change = 50 / 2
average rate of change = 25

Therefore, the average rate of change is 25 shares per minute.

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An answer option that could represent the quadratic equation, and why is: B.  x² + 2x - 35 = 0, the factors are (x + 7) and (x - 5) and (x + 7)(x - 5) = x² + 2x - 35.

In Mathematics and Geometry, the general form of a quadratic function can be modeled and represented by using the following quadratic equation;

y = ax² + bx + c

Where:

Next, we would solve the quadratic function by using the factors (zeros or roots) provided as follows;

y = (x + 7)(x - 5)

y = x² + 2x - 35

x² + 2x - 35 = 0

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Missing information:

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

The volume of the prism is 21 * 5 = 105 cubic feet.

To find the volume of a prism with a triangular base, you need to follow these steps:

1. Determine the area of the triangular base: Since the base is a right triangle with leg lengths of 6 feet and 7 feet, you can use the formula for the area of a right triangle: (1/2) * base * height. In this case, the area would be (1/2) * 6 * 7 = 21 square feet.

2. Multiply the area of the triangular base by the height of the prism: The prism is 5 feet tall, so the volume can be calculated by multiplying the area of the base (21 square feet) by the height (5 feet).

Thus, the volume of the prism is 21 * 5 = 105 cubic feet.

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Answer:

Baltimore Train: 1,020 mi/12 hr = 85 mph

Pennsylvania Train: 75 mph

So the correct answer is A.

The area of the playground include the following: 900 square yards.

In Mathematics and Geometry, the area of a regular polygon can be calculated by using this formula:

Area = (n × s × a)/2

Where:

In Mathematics and Geometry, the area of a parallelogram can be calculated by using the following formula:

Area of a parallelogram, A = base area × height

Area of playground, A = 45 × 20

Area of a playground, A = 900 square yards.

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The derivative of the vector function is: r'(t) = (-2t/(7-t^2)) i + (1/(2sqrt(13+t))) j - 36e^(9t) k

We are given a vector function r(t) = ln(7-t^2)i + sqrt(13+t)j – 4e^(9t)k, and we need to find its derivative r'(t).

The derivative of a vector function is obtained by differentiating each component of the vector function separately.

So, let's differentiate each component:

r(t) = ln(7-t^2)i + sqrt(13+t)j – 4e^(9t)k

r'(t) = (d/dt) ln(7-t^2) i + (d/dt) sqrt(13+t) j - (d/dt) 4e^(9t) k

Using the chain rule of differentiation, we have:

r'(t) = -2t/(7-t^2) i + 1/(2sqrt(13+t)) j - 36e^(9t) k

Therefore, the derivative of the vector function is:

r'(t) = (-2t/(7-t^2)) i + (1/(2sqrt(13+t))) j - 36e^(9t) k

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Answer:

I think it is k

Step-by-step explanation:

x+3(4x)

4x^2 + 12x

I am sorry if this is wrong

Since we have shown that all four angles formed by the lateral sides are right angles, and the opposite sides are parallel and congruent, we can conclude that the lateral sides are rectangles.

Yes, we can prove that the lateral sides are rectangles based on the given information.

Firstly, we can see that the two diagonals of the lateral sides are congruent (both measure 13 cm), which means that the opposite sides of the figure are parallel. This is because, in a rectangle, opposite sides are parallel and congruent.

Next, we can use the converse of the Pythagorean theorem to determine if the angles are right angles. The converse of the Pythagorean theorem states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

For each of the lateral sides of the figure, we can consider the two triangles formed by one of the diagonals and the adjacent sides. Applying the Pythagorean theorem, we can see that:

For the first lateral side, we have:

(5 cm)^2 + (12 cm)^2 = (13 cm)^2

Therefore, the angles between the 5 cm and 12 cm sides are right angles.

For the second lateral side, we have:

(5 cm)^2 + (12 cm)^2 = (13 cm)^2

Therefore, the angles between the 5 cm and 12 cm sides are also right angles.

Since we have shown that all four angles formed by the lateral sides are right angles, and the opposite sides are parallel and congruent, we can conclude that the lateral sides are rectangles.

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Answer:

Step-by-step explanation:

You want the segment lengths DA and DE of the hypotenuse in the triangle shown in the figure.

The radius to a point of tangency always makes a right angle with the tangent. This is a right triangle with legs 8 and 15, so you know from your knowledge of Pythagorean triples that the hypotenuse is 17.

  DA = 17

  DE = 17 -8 = 9

__

Additional comment

In case you haven't memorized a few of the useful Pythagorean triples, {3, 4, 5}, {5, 12, 13}, {7, 24, 25}, {8, 15, 17}, you can always figure the missing side length of a right triangle using the Pythagorean theorem.

It tells you the sum of the squares of the legs is the square of the hypotenuse:

  AC² +CD² = DA²

  8² +15² = DA²

  64 +225 = 289 = DA²

  DA = √289 = 17

Of course, AE is the radius of the circe, 8, so ...

  AE + DE = DA

  8 +DE = 17

  DE = 17 -8 = 9

Alternatively, you can solve this using the relation between tangents and secants. If the line DA is extended across the circle to intersect it again at X, then ...

  DC² = DE·DX

  15² = DE·DX = DE(DE +16) . . . . . . . EX is the diameter, twice the radius of 8

  DE² +16DE -225 = 0

  (DE +25)(DE -9) = 0 . . . . factor

  DE = 9  . . . . the positive solution

  DA = 9 +8 = 17

We like the Pythagorean theorem solution better, as the factors of the quadratic may not be obvious.

12+13 = 25, therefor the answer is correct

Therefore, the maximum value of f(x,y) inside the triangle is 80/9, which occurs along the line y = (-1/2)x + 4 at the point (8/3, 10/3), and the minimum value is -32, which occurs at the critical point (-8,4).

To find the maximum and minimum values of f(x,y) = xy on the region inside the triangle whose vertices are (6,2), (0,3), and (6,0), we use the method of Lagrange multipliers.

First, we need to find the critical points of f(x,y) subject to the constraint that (x,y) lies inside the triangle. We can express this constraint using the equations of the lines that form the sides of the triangle:

y = (-1/2)x + 4

y = (3/2)x

y = 0

Next, we set up the Lagrange multiplier equation:

∇f = λ∇g

where g(x,y) is the equation of the constraint, i.e., the triangle.

We have:

f(x,y) = xy

∇f = <y, x>

g(x,y) = y - (-1/2)x - 4 = 0

∇g = <-1/2, 1>

Setting ∇f = λ∇g, we get:

y = (-1/2)λ

x = λ

Substituting these into the constraint equation, we get:

(-1/2)λ - 4 = 0

Solving for λ, we get:

λ = -8

Substituting this into y = (-1/2)λ and x = λ, we get:

x = -8 and y = 4

Therefore, the only critical point of f(x,y) inside the triangle is (-8,4).

Next, we need to check the values of f(x,y) at the vertices and along the sides of the triangle.

At the vertices:

f(6,2) = 12

f(0,3) = 0

f(6,0) = 0

Along the line y = (3/2)x:

f(x, (3/2)x) = (3/2)x^2

Using the vertex (6,2) and the x-intercept (4/3, 2), we can see that the maximum value of (3/2)x^2 on this line occurs at x = 4. Therefore, the maximum value of f(x,y) along this line is:

f(4,6) = 24

Along the line y = (-1/2)x + 4:

f(x, (-1/2)x + 4) = (-1/2)x^2 + 4x

Using the vertex (6,2) and the x-intercept (8,0), we can see that the maximum value of (-1/2)x^2 + 4x on this line occurs at x = 8/3. Therefore, the maximum value of f(x,y) along this line is:

f(8/3,10/3) = 80/9

Finally, we need to check the values of f(x,y) at the critical point (-8,4). We have:

f(-8,4) = -32

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The smallest angle measures 15 degrees and the other non-right angle measures 75 degrees.

To find the measure of both angles in a right triangle with the given conditions, we will use the information provided:

Let x be the measure of the smallest angle. The problem states that the measure of one of the small angles is 30 less than 7 times the smallest angle, which can be written as:

Other non-right angle = 7x - 30

Since this is a right triangle, the sum of the two small angles must be 90 degrees (because the other angle is 90 degrees, and the sum of angles in a triangle is 180 degrees). So, we can set up the following equation:

x + (7x - 30) = 90

Now, solve for x:

8x - 30 = 90
8x = 120
x = 15

So, the smallest angle is 15 degrees. Now, we can find the measure of the other non-right angle:

Other non-right angle = 7x - 30 = 7(15) - 30 = 105 - 30 = 75 degrees

In summary, the smallest angle measures 15 degrees and the other non-right angle measures 75 degrees.

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Answer:Let the cost of an orange and that of a mango during the first market day be Ksh. x and Ksh. y respectively.

From the first market day:

30x + 12y = 936

From the second market day:

15(1.2x) + 20(3/4y) = 780

Simplifying the second equation:

18x + 15y = 780

(ii) Using matrix to find the cost of an orange and that of a mango in the first market day:

Rewriting the equations in matrix form:

|30 12| |x| |936|

|18 15| x |y| = |780|

Multiplying the matrices:

|30 12| |x| |936|

|18 15| x |y| = |780|

|30x + 12y| |936|

|18x + 15y| = |780|

Using matrix inversion:

| x | |15 -12| |936 12|

| y | = | -18 30| x |780 15|

|x| |270 12| |936 12|

| | = |-360 30| x |780 15|

|y|

Simplifying the matrix multiplication:

|x| |1194| |12|

| | = | 930| x |15|

|y|

Therefore, the cost of an orange in the first market day was Ksh. 39 and the cost of a mango in the first market day was Ksh. 63.

(iii) Calculation of the total amount of money realized for the sales:

On the second market day, Fatuma bought 15 oranges and 20 mangoes.

Cost of 15 oranges = 15(1.2x) = 18x

Cost of 20 mangoes = 20(3/4y) = 15y

Total cost of fruits bought on the second market day = 18x + 15y = 18(39) + 15(63) = Ksh. 1629

Profit earned on 15 oranges at 10% = 1.1(1.2x)(15) - (1.2x)(15) = 0.18x(15) = 2.7x

Profit earned on 20 mangoes at 15% = 1.15(3/4y)(20) - (3/4y)(20) = 0.15y(20) = 3y

Total profit earned = 2.7x + 3y

Total amount of money realized for the sales = Total cost + Total profit

= Ksh. 1629 + 2.7x + 3y.

Step-by-step explanation:

The hypothesis test suggests that there is not enough evidence to reject the claim made by the social media company that over 1 million people log onto their app daily, as the  t-value (-1.732) is less than the critical value (-2.429).

Null hypothesis, The true mean number of people who log onto the app daily is equal to or less than 1 million.

Alternative hypothesis, The true mean number of people who log onto the app daily is greater than 1 million.

Level of significance = 1%

We can use a one-sample t-test to test the hypothesis.

t = (X - μ) / (s / √n)

where X is the sample mean, μ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.

Substituting the values, we get

t = (998,946 - 1,000,000) / (23,876 / √65)

t = -1.732

Using a t-distribution table with 64 degrees of freedom and a one-tailed test at a 1% level of significance, the critical value is 2.429.

Since the calculated t-value (-1.732) is less than the critical value (-2.429), we fail to reject the null hypothesis. There is not enough evidence to support the claim that more than 1 million people log onto the app daily.

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The measure of the base angle x is approximately 81.54 degrees.

To get the measure of the base angle, we can use the fact that the sum of the angles in a triangle is always 180 degrees. Since this is an isosceles triangle, we know that the two base angles are congruent (they have the same measure).
Let's call the measure of each base angle y. Then we can set up an equation:
y + y + x = 180
Simplifying, we get:
2y + x = 180
Now we can use the fact that the legs of the triangle are congruent to find the measure of y. Since this is an isosceles triangle, we know that the two legs are congruent. This means we can use the Pythagorean theorem to find the length of the height, h, of the triangle:  h^2 = 9^2 - (12/2)^2
h^2 = 81 - 36
h^2 = 45
h = sqrt(45)
h = 6.71 (rounded to two decimal places)
Now we can use the definition of the tangent function to find y:
tan(y) = h / (12/2)
tan(y) = 6.71 / 6
tan(y) = 1.1183                                                                                                                                                                                     y = tan^-1(1.1183)
y = 49.23 degrees (rounded to two decimal places)
Finally, we can substitute this value of y into our equation to find x:
2y + x = 180
2(49.23) + x = 180
98.46 + x = 180
x = 81.54 degrees (rounded to two decimal places)
Therefore, the measure of the base angle x is approximately 81.54 degrees.

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Answer:

1st choice:  1/4(y - 10) = 2/3

Step-by-step explanation:

the "variable" is y

"is" means "=" (equals sign)

one fourth =  1/4

"difference of" means subtract

Answer:  1/4(y - 10) = 2/3

One point that would lie on the second line is (0,-4). Another  possible point on the 2nd line is (0, 12).

To find the equation of the first line, we can use the slope-intercept form:

y = mx + b

where m is the slope and b is the y-intercept. The slope of the line passing through (-3,8) and (1,9) can be found using the formula:

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

m = (9 - 8) / (1 - (-3))

m = 1/4

Using one of the points and the slope, we can find the y-intercept:

8 = (1/4)(-3) + b

b = 9

So the equation of the first line is:

y = (1/4)x + 9

To find the equation of the second line, we need to use the fact that it is perpendicular to the first line. The slopes of perpendicular lines are negative reciprocals, so the slope of the second line is:

m2 = -1/m1 = -1/(1/4) = -4

Using the point-slope form, we can write the equation of the second line:

y - 4 = -4(x + 2)

y - 4 = -4x - 8

y = -4x - 4

To find a point that lies on this line, we can plug in a value for x and solve for y. For example, if we let x = 0, then:

y = -4(0) - 4

y = -4

So the point (0,-4) lies on the second line.

Therefore, another point that would lie on the second line is (0,-4).

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Advertising cricket bats in a national newspaper provides Sachin's business with increased brand visibility and reach, while also posing a potential disadvantage due to high advertising costs.

One advantage and one disadvantage of Sachin's business advertising cricket bats in a national newspaper are as follows:

Advantage: Increased brand visibility and reach.

By advertising in a national newspaper, Sachin's business can reach a wider audience, creating greater brand awareness among potential customers. This increased visibility can contribute to the rapid increase in demand for cricket bats, ultimately leading to higher sales and profits for the business.

Disadvantage: High advertising cost.

National newspaper advertising can be quite expensive, especially for a business that advertises every two weeks. The high advertising costs might put financial pressure on Sachin's business, which could potentially affect other aspects of the business operations, such as product quality or employee wages. It's important for Sachin to weigh the benefits of national newspaper advertising against its costs to determine the most effective marketing strategy.

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The height of the lean-to on its vertical side is 8 feet.

To find the height of the lean-to on its vertical side, we need to use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, the vertical side is the hypotenuse, and the length and width are the other two sides.

So, we have:

[tex]height^2 = hypotenuse^2 - width^2[/tex]

We know the length of the roof (the hypotenuse) is 17 feet, and the width is 15 feet. So we can plug these values into the equation and solve for the height:

[tex]height^2 = 17^2 - 15^2\\height^2 = 289 - 225\\height^2 = 64\\height = 8[/tex]

Therefore, the height of the lean-to on its vertical side is 8 feet.

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To find the derivative of h(x) = 8x, we use the power rule, which states that the derivative of x^n is nx^(n-1). Applying this rule to h(x), we get h'(x) = 8.

To find the value of h'(4), we simply plug in x = 4 into our derivative expression: h'(4) = 8.

Therefore, the derivative of h(x) = 8x at x = 4 is h'(4) = 8.
To find the derivative of the function h(x) = 8x at x = 4, you can use the power rule for differentiation. The power rule states that if h(x) = x^n, then h'(x) = n * x^(n-1).

For h(x) = 8x, n = 1, so:

h'(x) = 1 * 8x^(1-1) = 8

Now, to find h'(4), just plug in x = 4:

h'(4) = 8

So, h'(4) = 8.

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A score of at least 183 is required to stay out of the bottom 1.5%. To find the minimum score required to receive the award, we need to determine the z-score corresponding to the top 3% of students.

Since the distribution is normal, we can use the standard normal distribution table to find the z-score. From the table, we find that the z-score corresponding to the top 3% is approximately 1.88.

Therefore, we can use the formula z = (x - μ) / σ, where μ = 400 and σ = 100, to find the minimum score required: 1.88 = (x - 400) / 100

Solving for x, we get: x = 1.88(100) + 400 = 488. Therefore, a score of at least 488 is required to receive the award.

To find the minimum score required to stay out of the bottom 1.5%, we need to determine the z-score corresponding to the bottom 1.5%.

From the standard normal distribution table, we find that the z-score corresponding to the bottom 1.5% is approximately -2.17. Therefore, we can use the same formula as before to find the minimum score required: -2.17 = (x - 400) / 100.

Solving for x, we get: x = -2.17(100) + 400 = 183. Therefore, a score of at least 183 is required to stay out of the bottom 1.5%.

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Using the quotient rule of logarithms, we have:

=log2 51.2 − log2 1.6

= [tex]log2 (51.2/1.6)[/tex]

Simplifying the numerator, we have:

[tex]log2(51.2/1.6) = log2(32)[/tex]

Using the fact that 32 = 2^5, we have:

log2 32 = log2 2^5 = 5

log2 51.2 − log2 1.6 = log2 (51.2/1.6) = log2 32 = 5

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The required answer is Jocelyn travels approximately 0.83 miles in 5 minutes.

Jocelyn's car tires are spinning at a rate of 120 revolutions per minute. If her car's tires are 28 inches in diameter, we can calculate the distance traveled in one revolution by finding the circumference of the tire:

Circumference = π x diameter
Circumference = 3.14 x 28 inches
Circumference ≈ 87.92 inches

So in one revolution, the car travels approximately 87.92 inches. To find out how many miles Jocelyn travels in 5 minutes, we need to multiply the number of revolutions in 5 minutes (which is 120 revolutions per minute x 5 minutes = 600 revolutions) by the distance traveled in one revolution (87.92 inches).

Distance traveled in 5 minutes = 600 revolutions x 87.92 inches/revolution

Distance traveled in 5 minutes = 52,752 inches

To convert inches to miles, we can use the conversion factor given: 1 mile = 63,360 inches.

Distance traveled in 5 minutes = 52,752 inches ÷ 63,360 inches/mile

Distance traveled in 5 minutes ≈ 0.83 miles

Therefore, Jocelyn travels approximately 0.83 miles in 5 minutes with her car tires spinning at a rate of 120 revolutions per minute. Rounded to the nearest hundredth, the answer is 0.83 miles.
To find out how many miles Jocelyn travels in 5 minutes, follow these steps:

1. Calculate the circumference of one tire: Circumference = Diameter × π.
Circumference = 28 inches × π ≈ 87.96 inches.

2. Determine the distance traveled in one revolution: One revolution covers the circumference of the tire, which is 87.96 inches.

3. Calculate the distance traveled in one minute: 120 revolutions per minute × 87.96 inches per revolution ≈ 10,555.2 inches per minute.

4. Determine the distance traveled in 5 minutes: 10,555.2 inches per minute × 5 minutes = 52,776 inches.

5. Convert the distance from inches to miles: 52,776 inches ÷ 63,360 inches per mile ≈ 0.83 miles.

So, Jocelyn travels approximately 0.83 miles in 5 minutes.

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The button's area is approximately 50.27 square centimeters.

To find the area of the button, we need to know the diameter of the button. We can find this by using the formula for circumference of a circle:

C = πd

where C is the circumference and d is the diameter.

Substituting the given value for C:

25.12 cm = πd

Solving for d:

d = 25.12 cm / π

d ≈ 8 cm

Now that we know the diameter, we can use the formula for area of a circle:

A = πr^2

where r is the radius (half the diameter).

Substituting the value for d:

r = d/2 = 4 cm

Substituting this value into the formula:

A = π(4 cm)^2

A ≈ 50.27 cm^2

Therefore, the button's area is approximately 50.27 square centimeters.

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The quotient of the expression (7¹⁸/7⁻⁹) expressed as a power of 7 is 7 ²⁷.

The quotient of the expression is calculated as follows;

When you divide two numbers with the same base, you subtract the exponents of the base. Using this rule, we can simplify the expression as follows;

7¹⁸ / 7⁻⁹

= 7 ⁽¹⁸ ⁻ ⁻⁹⁾

= 7 ⁽¹⁸ ⁺ ⁹⁾

= 7 ²⁷

Therefore, the quotient of the expression (7¹⁸/7⁻⁹) is 7 ²⁷.

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r2 = 0.523, which means that approximately 52.3% of the variance in the difference scores can be accounted for by the reduction in mental alertness after consuming tryptophan.

a. To test whether there is a significant reduction in mental alertness after consuming tryptophan versus before, we can use a paired samples t-test. The null hypothesis is that there is no difference in mental alertness scores before and after the meal, and the alternative hypothesis is that the scores are lower after the meal:

H0: μd = 0 (no difference)

Ha: μd < 0 (lower scores after the meal)

Here, μd is the mean difference score in mental alertness before and after the meal. We will use a one-tailed test with α = .05, since we are only interested in the possibility of lower scores after the meal.

The t-statistic for a paired samples t-test is calculated as:

t = (Md - μd) / (sd / sqrt(n))

Where Md is the mean difference score, μd is the hypothesized mean difference (in this case, 0), sd is the standard deviation of the difference scores, and n is the sample size.

We are given that Md = 14, and the standard deviation of the difference scores (sd) is:

sd = sqrt(SSd / (n - 1)) = sqrt(1152 / 8) = 12

Substituting these values, we get:

t = (14 - 0) / (12 / sqrt(9)) = 3.5

Using a one-tailed t-distribution table with 8 degrees of freedom and α = .05, the critical value is -1.86. Since our calculated t-value (3.5) is greater than the critical value, we reject the null hypothesis and conclude that there is a significant reduction in mental alertness after consuming tryptophan versus before.

b. To compute r2 to measure the size of the effect, we can use the formula:

r2 = t2 / (t2 + df)

Where t is the calculated t-value for the test, and df is the degrees of freedom, which is n-1 in this case.

Substituting the values , we get:

r2 = (3.5)2 / ((3.5)2 + 8) = 0.523

Therefore, r2 = 0.523, which means that approximately 52.3% of the variance in the difference scores can be accounted for by the reduction in mental alertness after consuming tryptophan.

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