Question

the area of rectangle abde is 270 square inches. the rectangle is rotated about a line that passes through points c and f.
which best describes the resulting three - dimensional shape?
○ a rectangular prism with an edge length of 9 inches
○ a rectangular prism with an edge length of 18 inches
○ a cylinder with a base radius of 9 inches
○ a cylinder with a base radius of 18 inches

Explanation:

Step 1: Determine the length of the rectangle

The area of a rectangle is given by \( A = l \times w \). We know the area \( A = 270 \) square inches and the width \( w = 15 \) inches. So, we can find the length \( l \) by rearranging the formula: \( l=\frac{A}{w} \). Substituting the values, we get \( l = \frac{270}{15}=18 \) inches.

Step 2: Analyze the rotation

When a rectangle is rotated about a line (in this case, the line through C and F, which is parallel to the side of length 15 inches and divides the length of 18 inches into two equal parts, so the distance from the line of rotation to the sides AB and DE is \( \frac{18}{2}=9 \) inches), the resulting three - dimensional shape is a cylinder. The radius of the base of the cylinder is the distance from the line of rotation to the side of the rectangle that is perpendicular to the line of rotation. Here, that distance is 9 inches, and the height of the cylinder will be equal to the length of the side of the rectangle that is parallel to the line of rotation (15 inches).

Step 3: Evaluate the options

• Option 1: A rectangular prism is not formed by rotating a rectangle about a line, so this is incorrect.
• Option 2: A rectangular prism is not formed by this rotation, so this is incorrect.
• Option 3: The resulting shape is a cylinder with a base radius of 9 inches (since the distance from the line of rotation to the side of the rectangle is 9 inches), this is correct.
• Option 4: The radius is 9 inches (not 18 inches) because the line of rotation is in the middle of the 18 - inch side, so this is incorrect.

Answer:

a cylinder with a base radius of 9 inches