A cylinder can be modeled as a rectangle rolled up around itself. The area of a rectangle is given by the formula A=bh. You want to find the area of the rectangle that surrounds the body of any given cylinder. The rectangle is the same height as the cylinder, and its base has the same length as the circumference of the circle that makes up the bottom and top of the cylinder. The circumference formula is C=2πr.

Question

Which of the following best builds a function for finding the area of a rectangle in a cylinder?
A. A=
h
2πr
​

B. A=
2πr
h
​

C. A=bh
2

D. A=2πrh

Answer 1

To find the area of the rectangle that surrounds a cylinder, we need to define the width (base) and height of the rectangle.

1. The height \( h \) of the rectangle is the same as the height of the cylinder.
2. The base \( b \) of the rectangle is equal to the circumference of the circle that forms the top and bottom of the cylinder, which is given by the formula \( C = 2\pi r \), where \( r \) is the radius of the circle.

The area \( A \) of the rectangle is calculated as: \[ A = \text{base} \times \text{height} = b \times h \]

Substituting the base \( b \) with the circumference \( C \): \[ A = (2\pi r) \times h \]

Therefore, the area \( A \) of the rectangle that surrounds the cylinder can be expressed as: \[ A = 2\pi rh \]

This corresponds to option D.

So, the best answer is: D. A = 2πrh