Question

question 10 of 10
which of the following formulas would find the surface area of a right cylinder where ( h ) is the height, ( r ) is the radius, and ( ba ) is the base area?
check all that apply.
a. ( 2pi r^{2} )
b. ( ba + 2pi rh )
c. ( 2pi r^{2}+2pi rh )
d. ( ba+pi r^{2} )
e. ( pi r^{2}+pi rh )

Explanation:

Step1: Recall the formula for the base area of a cylinder

The base area of a cylinder \(BA=\pi r^{2}\) (since the base is a circle with area formula \(A = \pi r^{2}\)).

Step2: Recall the formula for the lateral (curved) surface area of a cylinder

The lateral (curved) surface area of a cylinder is \(2\pi rh\).

Step3: Recall the formula for the total surface area of a cylinder

The total surface area of a cylinder \(S\) is the sum of the areas of the two bases and the lateral surface area. Since \(BA=\pi r^{2}\), the area of two bases is \(2BA = 2\pi r^{2}\). So \(S=2BA+2\pi rh\). Substituting \(BA = \pi r^{2}\), we get \(S = 2\pi r^{2}+2\pi rh\). Also, if we consider \(S=BA + BA+2\pi rh\), since \(BA\) is the base - area, we can write \(S=BA + 2\pi rh\) (where the first \(BA\) is for one base and \(2\pi rh\) is the lateral surface area and the second \(BA\) is accounted for in the general form when we express it in terms of the base - area formula).

• Option A: \(2\pi r^{2}\) only gives the area of the two circular bases, not the total surface area (missing the lateral surface area).
• Option D: \(BA+\pi r^{2}\) is incorrect. If \(BA=\pi r^{2}\), then \(BA+\pi r^{2}=2\pi r^{2}\) which is just the area of the two bases (missing the lateral surface area).
• Option E: \(\pi r^{2}+\pi rh\) is incorrect. It does not account for the full surface area (one base area and half of the lateral surface area conceptually wrong).

Answer:

B. \(BA + 2\pi rh\), C. \(2\pi r^{2}+2\pi rh\)