Question

A) A cylinder has a surface area of 42π cm . If it were dilated using a scale factor of
1
2
, what is
the surface area of the new cylinder? (1 point)
2
21π cm
2
84π cm
2
10.5π cm
2
13.5π cm
2

Answer 1

To find the new surface area after dilation, we first need to understand how the surface area of a solid changes with dilation. When an object is dilated by a scale factor \( k \), the surface area of the object changes by a factor of \( k^2 \).

Here, the original surface area of the cylinder is given as \( 42\pi \) cm², and the scale factor is \( \frac{1}{2} \).

1. Calculate the square of the scale factor:

\[
k^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}
\]

2. Now, multiply the original surface area by this factor:

\[
\text{New Surface Area} = \text{Original Surface Area} \times k^2 = 42\pi \times \frac{1}{4}
\]

\[
\text{New Surface Area} = \frac{42\pi}{4} = 10.5\pi \, \text{cm}^2
\]

Thus, the surface area of the new cylinder after dilation is \( 10.5\pi \, \text{cm}^2 \).

The correct answer is **10.5π cm²**.