A student says the two cylinders below have the same surface area. Explain the student’s error. (Use 3.14 for π.)

Question

A cylinder has a base diameter measuring 3 inches and a height measuring 10 inches. A second cylinder has a base diameter measuring 10 inches and a height measuring 3 inches.

Answer 1

The student is incorrect in assuming that the two cylinders have the same surface area. Surface area of a cylinder is calculated using the formula:

Surface Area = 2πr^2 + 2πrh

For the first cylinder with a base diameter of 3 inches and a height of 10 inches:
- radius (r) = diameter/2 = 3/2 = 1.5 inches
- Surface Area = 2(3.14)(1.5)^2 + 2(3.14)(1.5)(10) = 47.1 + 94.2 = 141.3 square inches

For the second cylinder with a base diameter of 10 inches and a height of 3 inches:
- radius (r) = diameter/2 = 10/2 = 5 inches
- Surface Area = 2(3.14)(5)^2 + 2(3.14)(5)(3) = 157 + 94.2 = 251.2 square inches

Therefore, the two cylinders do not have the same surface area. The second cylinder actually has a greater surface area than the first cylinder.

Answer 2

explain in four sentences why the student is incorrect

Answer 3

The student is incorrect because the surface area of a cylinder is not solely determined by its dimensions but also by its shape. The formula for calculating the surface area takes into account both the radius and height of the cylinder, not just the diameter. Therefore, the two cylinders with different dimensions will have different surface areas. In this case, the cylinder with a larger base diameter but shorter height will actually have a greater surface area than the cylinder with a smaller base diameter but taller height.