Question
Algebraic Reasoning and Linear Equations
1. A local charity is organizing a fundraising event. They plan to sell tickets for $15 each and expect to have x
attendees. Their goal is to raise at least $3000. Write an inequality to represent this situation, and solve it to
find the minimum number of attendees needed: ____ attendees.
2. The formula for the volume of a cylinder is V = πr2h , where V is the volume, r is the radius of the base, and
h is the height. If you want to find the height of a cylinder given its volume and radius, which of the following
correctly represents the rearranged formula?
A. h =
V
πr^2
C. h = V − πr^2
B. h =
V
πr
D. h =
V −πr^2
r
3. A rectangular swimming pool is being constructed in a community park. The length of the pool is 3 meters more
than twice its width. The perimeter of the pool is 50 meters. Set up and solve a system of equations to find the
dimensions of the pool. Show all your work and explain your reasoning for each step.
1. A local charity is organizing a fundraising event. They plan to sell tickets for $15 each and expect to have x
attendees. Their goal is to raise at least $3000. Write an inequality to represent this situation, and solve it to
find the minimum number of attendees needed: ____ attendees.
2. The formula for the volume of a cylinder is V = πr2h , where V is the volume, r is the radius of the base, and
h is the height. If you want to find the height of a cylinder given its volume and radius, which of the following
correctly represents the rearranged formula?
A. h =
V
πr^2
C. h = V − πr^2
B. h =
V
πr
D. h =
V −πr^2
r
3. A rectangular swimming pool is being constructed in a community park. The length of the pool is 3 meters more
than twice its width. The perimeter of the pool is 50 meters. Set up and solve a system of equations to find the
dimensions of the pool. Show all your work and explain your reasoning for each step.
Answers
GPT-5 mini
1) Let x = number of attendees. Each ticket is $15, so funds = 15x. They want at least $3000:
15x ≥ 3000
x ≥ 3000/15
x ≥ 200
Minimum number of attendees = 200.
2) Rearrange V = π r^2 h to solve for h: divide both sides by π r^2:
h = V / (π r^2).
Correct choice: A.
(Options B–D are incorrect: B divides by π r (missing one r), C subtracts instead of dividing, D has an incorrect subtraction/division order.)
3) Let W = width (m), L = length (m). Problem states:
L = 2W + 3. (length is 3 m more than twice the width)
Perimeter of a rectangle: P = 2(L + W) = 50, so
L + W = 25.
Substitute L from the first equation into the perimeter equation:
(2W + 3) + W = 25
3W + 3 = 25
3W = 22
W = 22/3 m ≈ 7.333... m
Then L = 2W + 3 = 2(22/3) + 3 = 44/3 + 3 = 44/3 + 9/3 = 53/3 m ≈ 17.666... m
Check: 2(L + W) = 2(53/3 + 22/3) = 2(75/3) = 2(25) = 50 m. OK.
Dimensions: width = 22/3 m (≈7.33 m), length = 53/3 m (≈17.67 m).
15x ≥ 3000
x ≥ 3000/15
x ≥ 200
Minimum number of attendees = 200.
2) Rearrange V = π r^2 h to solve for h: divide both sides by π r^2:
h = V / (π r^2).
Correct choice: A.
(Options B–D are incorrect: B divides by π r (missing one r), C subtracts instead of dividing, D has an incorrect subtraction/division order.)
3) Let W = width (m), L = length (m). Problem states:
L = 2W + 3. (length is 3 m more than twice the width)
Perimeter of a rectangle: P = 2(L + W) = 50, so
L + W = 25.
Substitute L from the first equation into the perimeter equation:
(2W + 3) + W = 25
3W + 3 = 25
3W = 22
W = 22/3 m ≈ 7.333... m
Then L = 2W + 3 = 2(22/3) + 3 = 44/3 + 3 = 44/3 + 9/3 = 53/3 m ≈ 17.666... m
Check: 2(L + W) = 2(53/3 + 22/3) = 2(75/3) = 2(25) = 50 m. OK.
Dimensions: width = 22/3 m (≈7.33 m), length = 53/3 m (≈17.67 m).