Question
The formula for the surface area of a cylinder is A = 2πr2 + 2πrh , where r is the radius and h is the height. In
this formula, the term 2πr2 represents the area of the ____ of the cylinder.
2. The cost C of producing x units of a product is given by the formula C = 500 + 25x + 0.1x2 . Which of the
following best describes the meaning of the term 0.1x2 in this context?
A. The fixed cost of
production
C. The additional cost
that increases at an
increasing rate as
more units are
produced
B. The variable cost
that increases linearly
with the number of
units
D. The profit gained
from selling x units
3. The formula for the volume of a cone is V =
1
3πr2h , where r is the radius of the base and h is the height of
the cone. Explain why doubling the radius would have a greater effect on the volume than doubling the height.
Use the structure of the formula to support your answer
this formula, the term 2πr2 represents the area of the ____ of the cylinder.
2. The cost C of producing x units of a product is given by the formula C = 500 + 25x + 0.1x2 . Which of the
following best describes the meaning of the term 0.1x2 in this context?
A. The fixed cost of
production
C. The additional cost
that increases at an
increasing rate as
more units are
produced
B. The variable cost
that increases linearly
with the number of
units
D. The profit gained
from selling x units
3. The formula for the volume of a cone is V =
1
3πr2h , where r is the radius of the base and h is the height of
the cone. Explain why doubling the radius would have a greater effect on the volume than doubling the height.
Use the structure of the formula to support your answer
Answers
GPT-5 mini
1. The two circular bases (the top and bottom) of the cylinder.
2. The additional cost that increases at an increasing rate as more units are produced. (The 0.1x^2 term is a quadratic variable cost — marginal cost grows with x.)
3. Because V = (1/3)πr^2h depends on r squared, changing r has a squared effect. If you double r (r → 2r), r^2 becomes (2r)^2 = 4r^2, so volume becomes 4 times as large. If you double h (h → 2h), volume only doubles. Thus doubling the radius increases volume more than doubling the height.
2. The additional cost that increases at an increasing rate as more units are produced. (The 0.1x^2 term is a quadratic variable cost — marginal cost grows with x.)
3. Because V = (1/3)πr^2h depends on r squared, changing r has a squared effect. If you double r (r → 2r), r^2 becomes (2r)^2 = 4r^2, so volume becomes 4 times as large. If you double h (h → 2h), volume only doubles. Thus doubling the radius increases volume more than doubling the height.