swerve wrote: ↑

Thu Feb 20, 2020 9:27 am

2019-04-30_1303.png

A rectangular label of width x has been wrapped around the cylinder above, encircling the cylinder without overlap. If the radius of the cylinder is 6, and the label has the same area as the base of the cylinder, then what is the value of x?

A. 3
B. 5
C. 6
D. \(6\pi\)
E. \(9\pi\)

The OA is A

Source: Princeton Review

If we UNWRAP the label, we see that the label is a RECTANGLE
The height of the rectangle is x, and the width of the rectangle is equal to the circumference of the cylinder

Circumference = 2πr
So, in the above diagram, circumference = (2)(π)(6) = 12π
So the area of the rectangular label = (base)(height) = (12π)(x)

GIVEN: The label has the same area as the base of the cylinder
Area of circle = πr²
So the area of the cylinder's base = π(6²) = 36π

Since the two areas are equal we can write: 36π = (12π)(x)

Divide both sides of the equation by 12π to get: 3 = x

Answer: A

Cheers,
Brent