Mathematics/g12
nsc
march 2021
question1
a group of workers is erecting a fence around a nature reserve. they store their tools in a shed at
the entrance to the reserve. each day they collect their tools and erect 0,8km of new fence. they
then lock up their tools in the shed and return the next day.
1.1 if the fence takes 40 days to erect, how far would the workers have travelled in total?
(4)​

Answer:

Step-by-step explanation:

AngleSideSide because it's bad

but also, if you had an angle, a side and a side

For your example: let's say CD≅AS

You could change the angle of S or D and the parameters of the triangle would still be true.   Because you can change something and still have AngleSideSide be true, would make them not congruent any more.

The correct expression is 4,000 times. 1. 4 to the sixth power.

The formula for the future value of an investment with annual compounding interest is:

A = P(1 + r)ⁿ

A = future value

P = principal amount

r = annual interest rate expressed as a decimal

n = number of years.

In this case, Jaleesa deposited $4,000 at an annual interest rate of 4% (0.04 as a decimal) and the investment is compounded annually for 6 years. So the expression that can be used to find the value of her investment at the end of 6 years is:

A = 4,000(1 + 0.04)⁶

Simplifying the expression, we get:

A = 4,000(1.04)⁶

Therefore, the correct expression is 4,000 times. 1. 4 to the sixth power.

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The probability of hitting the bulls eye (center circle) when we shoot an arrow with radius of the bulls eye is 2ft and the radius of the target is 6ft is 11.1%.

Assuming that we are shooting randomly at the target and the arrow can land anywhere within the target area, the probability of hitting the bull's eye (center circle) can be calculated as the ratio of the area of the bull's eye to the area of the entire target.

The area of the bull's eye can be calculated as follows:

Area of bull's eye = π x (radius of bull's eye)²
Area of bull's eye = 3.14 x 2²
Area of bull's eye = 12.56 square feet

The area of the entire target can be calculated as follows:

Area of target = π x (radius of target)²
Area of target = 3.14 x 6²
Area of target = 113.04 square feet

Therefore, the probability of hitting the bull's eye can be calculated as:

Probability of hitting bull's eye = Area of bull's eye / Area of target
Probability of hitting bulls eye = 12.56 / 113.04
Probability of hitting bulls eye = 0.111 or approximately 11.1%

So, the probability of hitting the bull's eye (center circle) when we shoot an arrow randomly at the target is approximately 11.1%.

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The evaluated probability of randomly choosing a  participant dreams in black and white or color is 0.13, under the condition that the given Students are passing from psychology surveyed 200 of their fellow students regarding their dreams.

Probability means the possible chances of an event occurring in a particular time frame. It is a considered a branch of mathematics that deals with the occurrence of a random event.

The value is presented from zero to one. Probability has been induced in math to predict how prone are the events going to occur. The meaning of probability is basically the express  something that is likely to happen.

Black Probability = 4/200=0.02

White Probability = 12/200=0.06

Probability of all other color = 10/200=0.05

So, probability of randomly choosing a participant dreams in black and white or color =0.02+0.06+0.05=0.13

Therefore, the probability of randomly selecting participant dreams in black and white or color = 0.13

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The complete question is

Students majoring in psychology surveyed 200 of their fellow students about their dreams. The results of the survey are shown in the Venn diagram. Let B be the event that the participant dreams in black and white and let C be the event that the participant dreams in color.

What is the probability that a randomly selected participant dreams in black and white or color?

Given that quadrilateral PQRS is a parallelogram, you can prove that it is also a rectangle by A: Use the distance formula to find the length of both diagonals to see if they are congruent and B: Find the slopes of all sides to determine if the angles are right angles. Therefore, the correct option is C: C. Both A and B are valid.

To prove that PQRS is a rectangle, we need to show that all angles are right angles.

Option A: Using the distance formula, we can find the lengths of both diagonals, PR and QS. If PR and QS are congruent, then we know that opposite sides of the parallelogram are congruent and parallel (since PQRS is a parallelogram). If we can also show that PR and QS intersect at a 90-degree angle, then we have proven that PQRS is a rectangle.

Option B: Finding the slopes of all sides can help us determine if the angles are right angles. If the product of the slopes of adjacent sides is -1, then we know that the sides are perpendicular (since the slope of a line perpendicular to another line is the negative reciprocal of its slope). If we can show that all adjacent sides have slopes that multiply to -1, then we have proven that PQRS is a rectangle.

Both options A and B can be used to prove that PQRS is a rectangle, so the correct answer is C.

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Using the Mean Value Theorem, we have proven that √(6a+3) < a + 2 for all a > 1.

To prove √(6a+3) <a + 2 for all a > 1 using the Mean Value Theorem, we will begin by defining a function f(x) as:

f(x) = √(6x+3)

We can see that f(x) is a continuous and differentiable function for all x > -1/2.

Now, let's choose two values of a, such that a > 1 and b = a + h, where h is a positive number. By the Mean Value Theorem, there exists a value c between a and b such that

f(b) - f(a) = f'(c)(b-a)

where f'(c) is the derivative of f(x) evaluated at c.

Now, let's evaluate the derivative of f(x) as:

f'(x) = 3/(√(6x+3))

Thus, we can write

f(b) - f(a) = f'(c)(b-a)

√(6(a+h)+3) - √(6a+3) = f'(c)h

Dividing both sides by h and taking the limit as h → 0, we get

lim h→0 (√(6(a+h)+3) - √(6a+3))/h = f'(a)

Now, we can evaluate the limit on the left-hand side using L'Hopital's rule

lim h→0 (√(6(a+h)+3) - √(6a+3))/h = lim h→0 [3/(√(6(a+h)+3)) - 3/(√(6a+3))] = 3/(2√(6a+3))

Therefore, we have

f'(a) = 3/(2√(6a+3))

Now, we can use this value to rewrite the inequality as

√(6a+3) - (a + 2) < 0

Multiplying both sides by 2√(6a+3) and simplifying, we get

3 < 4a + 2√(6a+3)

Subtracting 4a from both sides and squaring, we get

9 < 16a^2 + 16a + 24a + 12

Simplifying, we get

0 < 16a^2 + 40a + 3

This inequality holds for all a > 1, so we have proved that

√(6a+3) < a + 2 for all a > 1.

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The given question is incomplete, the complete question is:

Use Mean value theorem to prove √(6a+3) <a + 2 for all a > 1. Using methods other than the Mean Value Theorem will yield

6 gallons of gas were used to travel 210 miles.

A4. 1) To find the amount of gas used to travel 210 miles, we can rearrange the given equation m=35g to solve for g:

g = m/35

Substituting m=210, we get:

g = 210/35 = 6

6 gallons of gas were used to travel 210 miles.

A4. 2) To find the inverse function, we need to solve the given equation for g in terms of m:

m = 35g

Divide both sides by 35:

g = m/35

So the inverse function is:

g(m) = m/35

A4. 3) Similar to A4.2, we need to solve the given equation m=hg for g in terms of m:

m = hg

Divide both sides by h:

g = m/h

Now, we can write the inverse function as:

g(m) = m/h\

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The surface area of the rectangular prism in the image above is determined as:

236 square meters.

The prism given above in the image is a rectangular prism. The formula for finding the surface area of the prism is given as:

surface area of the prism = 2(lh + lw + hw), where:

h is the height

w is the width

l is the length of the prism.

Given the following:

length (l) = 6 m

width (w) = 5 m

height (h) = 8 m

Plug in the values:

Surface area of the prism = 2·(5·6 + 8·6 + 8·5) = 236 square meters.

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The estimated areas of each curve are listed below:

Case 1: A = 12.75

Case 2: A = 12.5

In this problem we must estimate the area above the x-axis and under a curve by using sums of rectangles and triangles according to the following expression:

A = {∑ [MIN (f(xₙ₋₁), f(xₙ))] + 0.5 · ∑ [MAX (f(xₙ₋₁), f(xₙ)) - MIN (f(xₙ₋₁), f(xₙ))]} · Δx, for n = {1, 2, 3, ..., N}

Where:

Case 1

A = (3.5 + 3.5 + 1.5) · 1 + 0.5 · (3.5 + 0.75 + 0.75 + 2 + 1.5) · 1

A = 8.5 + 0.5 · 8.5

A = 12.75

Case 2

A = (1 + 3 + 4) · 1 + 0.5 · (1 + 2 + 1.5 + 0.5 + 4) · 1

A = 8 + 4.5

A = 12.5

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The maximum height of the arrow is 105 feet. To find the maximum height of the arrow, we need to determine the vertex of the quadratic function h = -16[tex]t^{2}[/tex] + 80t + 25.

The vertex is the highest point on the graph of the function, which represents the maximum height of the arrow.

To find the t-value at the vertex, we use the formula t = -b/2a, where a = -16 and b = 80. Plugging these values into the formula gives us t = -80/(2(-16)) = 2.5 seconds.

To find the maximum height, we plug t = 2.5 into the equation to get h = -16[tex](2.5)^{2}[/tex] + 80(2.5) + 25 = 105 feet. Therefore, the maximum height of the arrow is 105 feet.

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The height of the wooden post cap is approximately 11.34 centimeters.

To find the height of the wooden post cap, we need to use the formula for the volume of a square pyramid which is V = (1/3) Bh, where B is the area of the base and h is the height. Since we know that each post cap has a volume of 94.5 cubic centimeters, we can plug this into the formula to get:

94.5 = (1/3) B h

We also know that the base of the pyramid is a square, so the area of the base is equal to s², where s is the length of one side. Since we don't know the length of the side or the height of the pyramid, we need to use another piece of information. Let's assume that the fence posts are all the same height and that the caps fit snugly on top of them. If we measure the height of the fence post, we can use this to find the height of the cap.

Let's say that the height of the fence post is 10 centimeters. We can use this to find the area of the base:

B = s² = (10/2)² = 25 square centimeters

Now we can plug this into the formula:

94.5 = (1/3) (25) h

Simplifying this equation, we get:

h = (94.5 * 3) / 25 = 11.34 centimeters

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The range of costs that Kenny could spend on his last gift would be any amount less than or equal to $33.03. So the range would be: $0 ≤ x ≤ $33.03

To determine the range of costs Kenny could spend on the last gift, we'll first calculate the total amount he has spent so far and subtract that from the $150 he started with.

Kenny has spent $68.99 (watch) + $38.99 (arrows) + $8.99 (lunch) = $116.97.

Now, let x represent the cost of the last gift. The inequality to represent the situation is:

116.97 + x ≤ 150

To find the range of costs for the last gift, subtract 116.97 from both sides of the inequality:

x ≤ 150 - 116.97
x ≤ 33.03

So, the range of costs Kenny could spend on the last gift is $0 to $33.03.

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The given exponential function satisfies the property of increasing by a factor of 3 between any two points separated by x2−x1=1.

An exponential function is a mathematical function of the form f(x) = a^x, where a is a positive constant and x is any real number. The base a is typically a number greater than 1, and the function grows or decays rapidly depending on whether a is greater than or less than 1.

Exponential functions are commonly used to model processes that exhibit exponential growth or decay, such as population growth, radioactive decay, and compound interest. They also arise in various areas of mathematics and science, including calculus, probability theory, and physics.

We are given the exponential function [tex]y=3^x.[/tex]

Let x1 be any value of x, then the corresponding y-value is [tex]y1=3^{(x_1)[/tex]

Let x2=x1+1 be the next value of x, then the corresponding y-value is [tex]y2=3^x2=3^(x1+1)=3*3^x1.[/tex]

So, we can see that y2 is 3 times y1, which means the y-values increase by a factor of 3 between any two points separated by x2−x1=1.

Therefore, the given exponential function satisfies the property of increasing by a factor of 3 between any two points separated by x2−x1=1.

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The measure of ∠O in ΔMNO is approximately 41.5°.

To find the measure of ∠O, we can use the Law of Cosines in ΔMNO, with sides M = 540 inches, N = 330 inches, and O = 600 inches. The Law of Cosines states:

O² = M² + N² - 2MN * cos(∠O)

Rearrange the equation to solve for cos(∠O):

cos(∠O) = (M² + N² - O²) / (2MN)

Substitute the values:

cos(∠O) = (540² + 330² - 600²) / (2 * 540 * 330)

cos(∠O) ≈ -0.7944

Now, find the angle using the inverse cosine function:

∠O ≈ arccos(-0.7944) ≈ 141.5°

Since ∠O is an obtuse angle, we need to find its supplement to the nearest 10th:

180° - 141.5° ≈ 38.5°

Thus, the measure of ∠O is approximately 38.5°.

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Based on the following  observations, we can write the equation of the rational function as: f(x) = (x + 1)/(x - 1)

A rational function is a type of mathematical function that is defined as the ratio of two polynomial functions.

In other words, it is a function that can be expressed as f(x) = p(x)/q(x), where p(x) and q(x) are polynomials and q(x) is not the zero polynomial.

To find the equation of the rational function represented by the given graph, we need to analyze the behavior of the graph and identify its key features. given below are the steps:

Look at the behavior of graph as x approaches infinity and negative infinity. The graph appears to have horizontal asymptotes at y = -1 and y = 1. This suggests that the function has a degree of 1 in both the numerator and denominator.

Identify any vertical asymptotes. The graph have vertical asymptote at x = 1. This suggests that the denominator of the function has a factor of (x - 1).

Look for any x-intercepts or y-intercepts.The graph's x-intercept and y-intercept are both at x = -1 and 1, respectively. This suggests that the numerator of the function has a factor of (x + 1) and that the function has a constant term of 1 in the numerator.

This function has a degree of 1 in both the numerator and denominator, a vertical asymptote at x = 1, and horizontal asymptotes at y = -1 and y = 1. It also has an x-intercept at x = -1 and a y-intercept at y = 1, which match the features of the graph given.

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To evaluate the limit using L'Hospital's rule, we need to take the derivative of both the numerator and denominator separately until we get a determinate form. We have:

lim (e^x + 2x - 1) / (2x)

Taking the derivative of the numerator:

lim (e^x + 2) / 2

Taking the derivative of the denominator:

lim 2

Since we now have a determinate form, we can evaluate the limit by plugging in the value of x. We get:

(e^x + 2) / 2

As x approaches infinity, e^x also approaches infinity, so the limit diverges to positive infinity. Therefore, the limit does not exist.

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A positive integer is 40 more than 29 times another. Their product is 10116 these two integers are 19 and 571.

In mathematics, an integer is a whole number that can be positive, negative, or zero. Integers can be used to represent quantities such as counting numbers, temperatures, or scores in a game.

Let's call the two integers x and y, where x is the larger integer and y is the smaller integer.

From the problem, we know that:

x = 29y + 40 (equation 1)

xy = 10116 (equation 2)

We can substitute equation 1 into equation 2 to get:

(29y + 40)y = 10116

Expanding and simplifying:

29[tex]y^{2}[/tex]+ 40y - 10116 = 0

We can use the quadratic formula to solve for y:

y = (-40 ± √([tex]40^{2}[/tex] - 429(-10116))) / (2×29)

y = (-40 ± √308576) / 58

y ≈ 18.95 or y ≈ -12.3

Since y is a positive integer, we can round up to 19.

Now we can use equation 1 to find x:

x = 29y + 40

x = 29(19) + 40

x = 571

Therefore, the two integers are 19 and 571.

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The direction angles of the vector u = (-1, 9, -6) are approximately -6.1°, 64.8°, and -75.2° for the x, y, and z axes, respectively.

The direction angles of a vector are the angles that the vector makes with the positive x, y, and z axes. To find the direction angles of the vector u = (-1, 9, -6), we can use the formulas:

cosθx = u_x/||u||, cosθy = u_y/||u||, and cosθz = u_z/||u||

where θx, θy, and θz are the angles that u makes with the x, y, and z axes, respectively, and ||u|| is the magnitude of u, given by:

||u|| = √(u_x² + u_y² + u_z²)

Substituting the values of u, we have:

||u|| = √((-1)² + 9² + (-6)²) = √118

cosθx = -1/√118 ≈ -0.183, cosθy = 9/√118 ≈ 0.551, and cosθz = -6/√118 ≈ -0.366

Taking the inverse cosine of each of these values, we get:

θx ≈ -6.1°, θy ≈ 64.8°, and θz ≈ -75.2°

Therefore, the direction angles of the vector u are approximately -6.1°, 64.8°, and -75.2° for the x, y, and z axes, respectively.

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The number of left-handed students in the first trial of the simulation can be found by following the above steps and counting the occurrences of two-digit numbers within the 00-14 range on line 1 of the random-number table.

To find out how many left-handed students occur in the first trial of the simulation, you'll need to follow these steps,
1. Identify the probability range for left-handed students, which is 00-14 as you've mentioned.
2. Start on line 1 of the random-number table.
3. Read each two-digit number on the line and check if it falls within the range 00-14.
4. Count the number of times a number within the 00-14 range appears in the first 10 two-digit numbers (since you're selecting a random sample of 10 students).
5. The count of numbers within the 00-14 range represents the number of left-handed students in the first trial of the simulation.

By doing the aforementioned processes and counting the occurrences of two-digit numbers between the ranges of 00 and 14 on line 1 of the random-number table, it is possible to determine the number of left-handed pupils in the simulation's first trial.

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To determine this family's annual medical aid tax credit, we need to consider their medical aid expenses for the year. Medical aid expenses are expenses related to medical services that are not covered by the government or medical insurance.

This family can claim a tax credit of up to R310 per month for the main member and the first dependent, and R209 per month for each additional dependent. This tax credit is only applicable to registered medical schemes and is deducted from the tax payable.

In addition to medical aid expenses, this family will also need to consider the 15% VAT payable on all goods and services, except for sanitary pads, fresh produce, and a few other staple food items. This means that if this family spends R10,000 on goods and services, they will need to pay an additional R1,500 in VAT.

However, the good news is that they won't have to pay VAT on their fresh produce and staple food items. This will help to reduce their overall expenditure on food, which is an essential expense for every family.

In conclusion, while this family will need to pay VAT on most goods and services, they can claim a tax credit for their medical aid expenses. Additionally, they won't have to pay VAT on fresh produce and staple food items, which will help to reduce their overall food expenditure.

By carefully managing their expenses and taking advantage of tax credits and exemptions, this family can ensure that they are able to provide for their essential needs while also managing their financial obligations.

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Answer:

Using the Distance Formula:

EG =√((-a - b)^2) + (0 - c)^2)

=√((a + b)^2 + c^2)

FH =√((-b - a)^2 + (c - 0)^2)

=√((a + b)^2 + c^2)

So EG = FH.

In the given diagram, the value of x in the right triangle is approximately 14.7

From the question, we are to calculate the value of x in the given diagram.

In the given diagram, we have two right triangles.

First, we will calculate the value of the side joining the two triangles.

Let the side joining the two triangles be d

Thus,

From the Pythagorean theorem, we can write that

d² = a² + b²

d² = 10² + 6²

d² = 100 + 36

d² = 136

Now, in the other triangle

Also, using the Pythagorean theorem, we can write that

x² = c² + d²

x² = 9² + 136

x² = 81 + 136

x² = 217

x = √217

x = 14.7309

x ≈ 14.7

Hence, the value of x is approximately 14.7

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The event that has a probability of 0 is selecting a yellow marble

Other probabilities are listed below

The items in the jar are given as

6 black, 4 brown, 8 white, and 2 blue.

These items are the sample space of this event

Hence, the sample space is 6 black, 4 brown, 8 white, and 2 blue.

For the probability Julie will draw a white marble, we have

P(White) = White/Total

So, we have

P(White) = 8/20

Simplify

P(White) = 2/5

For the event that is more likely to happen, we have

P(black marble) = 6/20

P(brown or blue marble) = (4 + 2)/20

P(brown or blue marble) = 6/20

The probabilities are equal

So, both events have equal likelihood

The event that has a probability of 0 could be the probability of selecting a yellow marble

This is because the jar has no yellow marble

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The amount in pounds of kale Lori bought is 1 1/2 pounds.

To find out how many pounds of kale Lori bought, you need to multiply the total weight of the vegetables by the fraction that represents the proportion of kale.

In this case, you can calculate the amount of kale as follows:

(7 1/2) * (1/5)

First, convert the mixed number 7 1/2 to an improper fraction:

(7 * 2 + 1) / 2 = 15/2

Now multiply the two fractions:

(15/2) * (1/5)

Multiply the numerators together and the denominators together:

(15 * 1) / (2 * 5) = 15 / 10

Now, simplify the fraction:

15 ÷ 5 / 10 ÷ 5 = 3 / 2

So, Lori bought 3/2 or 1 1/2 pounds of kale from the grocery store. This means that out of the total 7 1/2 pounds of vegetables she purchased, 1 1/2 pounds were kale.

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(a) To minimize the area of the right triangle formed by the cable, ground, and building, we need to minimize the length of the cable. To do this, we can use the Pythagorean theorem:

c^2 = a^2 + b^2

where c is the length of the cable, a is the distance from the guidepost to the point where the cable is attached to the building, and b is the distance from that point to the ground.

Since we want to minimize c, we can differentiate the equation with respect to a and set the derivative equal to zero:

dc/da = 2a/c = 0

Solving for a, we get a = c/2. This means that the point where the cable is attached to the building should be halfway up the building, or 10 feet high.

(b) To minimize the length of the cable, we can use the principle of least action, which states that the path taken by the cable is the one that minimizes the integral of the tension along the cable.

Assuming that the tension in the cable is constant, we can use the law of sines to find the angle θ:

sin θ / 20 = sin (90° - θ) / c

where c is the length of the cable.

We want to minimize c, so we can differentiate the equation with respect to θ and set the derivative equal to zero:

d(c)/d(θ) = -20cos(θ) / sin^2(θ) + cos(θ) / sin(θ) * dc/d(θ) = 0

Solving for dc/d(θ), we get:

dc/d(θ) = 20c * tan(θ)

Substituting this into the original equation, we get:

-20cos(θ) / sin^2(θ) + cos(θ) / sin(θ) * 20c * tan(θ) = 0

Simplifying, we get:

cos(θ) / sin(θ) = tan(θ)

Solving for θ, we get:

θ = 45°

Therefore, to minimize the length of the cable, it should make an angle of 45° with the face of the building.

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Answer:

Ben is 31 cm taller

Step-by-step explanation:

1090mm is 109 cm

Subtract Fareeds height from Bens:

140-109=31

Hope this helps!

Answer:

Based on the given information, we can conclude that the graph represents a quadratic function. The vertex of the parabola is located at (0, 9) and the function passes through several points including (-4, -7), (-3, 0), (3, 0), and (4, -7).

To find the equation of the function, we need to determine the value of "a" in the equation f(x) = ax^2 + bx + c. Since the vertex is located at (0, 9), we know that the x-coordinate of the vertex is 0. Therefore, we can use the vertex form of the equation, which is f(x) = a(x - 0)^2 + 9, or simply f(x) = ax^2 + 9.

Next, we can use one of the given points to solve for "a". Let's use the point (-3, 0).

0 = a(-3)^2 + 9

0 = 9a - 9

9 = 9a

a = 1

Therefore, the value of "a" in the equation of the function is B. 1.

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Step-by-step explanation:

A, 7^(x+3) = 823, 543

7^(x+3) = 823, 543

7^(x+3) = 7^7 ....... write in the power form which it's base must be 7 in order to equalify the power

x+3 = 7

x = 4

B, 4^ -4x = 4^-8

4 ^ -4x = 1/65, 536

4^ -4x = 1/4^8 ........ write in the power form

4^ -4x = 4^-8

-4x = -8 ..... write equality among the power cause it's base is same

x=2

C, 1/(6^(x-5) ) = 1296

1/(6^(x-5) ) = 1296

6^-(x-5) = 1296

6^-(x-5) = 6^4........ write in the power form

-(x-5) = 4

-x + 5 = 4

-x = -11

x = 1

D, 1/3^x+7 = 1/243

1/3^x+7 = 1/243

3 ^ -(x+7) = 3^-5 .... write in the power form

-(x+7) = -5

-x-7 = -5

-x = 2

x= -2

The volume of the largest right cylinder that fits in a sphere of radius 4 is 128π cubic units.

To find the volume, we need to understand that the cylinder that fits inside a sphere will have its height (h) equal to the diameter of the sphere (2r), and the cylinder's radius (r') will also be equal to the sphere's radius (r).

We can use the formula for the volume of a cylinder: V = π[tex]r^2^h[/tex], where π is pi (approximately 3.14), r is the radius, and h is the height.

Since the cylinder's height is equal to the sphere's diameter, which is 2r, the height of the cylinder is 2r. Therefore, we can write the volume of the cylinder as:

V = πr²(2r)

Simplifying this expression, we get:

V = 2π[tex]r^3[/tex]

To find the maximum volume of the cylinder that fits inside a sphere of radius 4, we need to maximize the volume by finding the maximum value of r. Since the radius of the cylinder is equal to the radius of the sphere, we have:

r = 4

Substituting this value into the formula for the volume of the cylinder, we get:

V = 2π[tex](4)^3[/tex]

V = 128π

Therefore, the volume of the largest right cylinder that fits in a sphere of radius 4 is 128π cubic units.

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