Question

the smaller cylinder has a radius of 3 inches and 12 inches. if the height of the larger cylinder is 24 inches, what is the surface area of the smaller cylinder? round your answer to the nearest tenth. in² need help?

Explanation:

Step1: Determine the scale factor

The radii of the two cylinders are 3 inches (smaller) and 12 inches (larger). The scale factor \( k \) is the ratio of the smaller radius to the larger radius: \( k=\frac{3}{12}=\frac{1}{4} \).

Step2: Find the height of the smaller cylinder

The height of the larger cylinder is 24 inches. Since the cylinders are similar, the ratio of their heights is equal to the scale factor. Let \( h \) be the height of the smaller cylinder. Then \( \frac{h}{24}=\frac{1}{4} \), so \( h = 24\times\frac{1}{4}=6 \) inches.

Step3: Calculate the surface area of a cylinder

The formula for the surface area of a cylinder is \( SA = 2\pi r^2+ 2\pi r h \), where \( r \) is the radius and \( h \) is the height.

For the smaller cylinder, \( r = 3 \) inches and \( h = 6 \) inches.

First, calculate the area of the two circular bases: \( 2\pi r^2=2\pi(3)^2 = 18\pi \).

Then, calculate the lateral (curved) surface area: \( 2\pi r h=2\pi(3)(6)=36\pi \).

Step4: Sum the areas

Add the two parts together: \( SA=18\pi + 36\pi=54\pi \).

Now, calculate the numerical value: \( 54\pi\approx54\times3.1416\approx169.6 \) (rounded to the nearest tenth).

Answer:

\( 169.6 \) square inches