The cylinders shown below are similar.
Two cylinders are shown side by side. The first cylinder has a height of 3 inches and a radius of 1 inch. The second cylinder has a height of 9 inches and a radius of 3 inches.
What is the ratio of their surface areas?
A. 1/27
B. 3/27
C. 1/9
D. 1/3
The formula for the surface area of a cylinder is $2\pi rh+2\pi r^2$, where $r$ is the radius and $h$ is the height. Since the two cylinders are similar, their corresponding dimensions have the same ratio. Therefore, the ratio of their surface areas is equal to the square of the ratio of their radii, or $\left(\frac{1}{3}\right)^2=\frac{1}{9}$. Answer $\boxed{\textbf{(C)}}$.
To find the ratio of their surface areas, we need to calculate the surface area of each cylinder.
The surface area of a cylinder is given by the formula:
A = 2πr² + 2πrh
Let's calculate the surface area for each cylinder:
Cylinder 1:
Height (h₁) = 3 inches
Radius (r₁) = 1 inch
A₁ = 2π(1)² + 2π(1)(3)
A₁ = 2π + 6π
A₁ = 8π
Cylinder 2:
Height (h₂) = 9 inches
Radius (r₂) = 3 inches
A₂ = 2π(3)² + 2π(3)(9)
A₂ = 18π + 54π
A₂ = 72π
Now, we can find the ratio of the surface areas by dividing the surface area of Cylinder 2 (A₂) by the surface area of Cylinder 1 (A₁):
Ratio = A₂ / A₁
Ratio = (72π) / (8π)
Ratio = 9
Therefore, the ratio of their surface areas is 9.
The correct option is:
C. 1/9