A student says the two cylinders below have the same surface area: cylinder one has diameter 3 inches and height 10 inches; cylinder two has diameter 10 and 3 height; Explain the student’s error
The student has made an error in assuming that the surface area of a cylinder depends only on its dimensions. In fact, the surface area of a cylinder is calculated using the formula:
Surface Area = 2πr^2 + 2πrh
where r is the radius of the base and h is the height of the cylinder.
In this case, for cylinder one, the radius of the base is 3/2 = 1.5 inches and the height is 10 inches. Therefore, the surface area of cylinder one is:
Surface Area = 2π(1.5)^2 + 2π(1.5)(10)
Surface Area = 2π(2.25) + 2π(15)
Surface Area = 4.5π + 30π
Surface Area = 34.5π square inches
For cylinder two, the radius of the base is 10/2 = 5 inches and the height is 3 inches. Therefore, the surface area of cylinder two is:
Surface Area = 2π(5)^2 + 2π(5)(3)
Surface Area = 2π(25) + 2π(15)
Surface Area = 50π + 30π
Surface Area = 80π square inches
As we can see, the surface areas of the two cylinders are different. It is important to use the correct formula and dimensions when calculating the surface area of a cylinder.