a cylinder has a surface area of 54pi cm^2. if it were dilated using the scale factor of 1/2, what is the surface area of the new cylinder

Answer 1

To find the surface area of the new cylinder after applying a scale factor of \( \frac{1}{2} \), we can use the following property of surface areas when scaling:

When an object is dilated by a scale factor \( k \), the surface area scales by the factor \( k^2 \).

Given that the original surface area of the cylinder is \( 54\pi , \text{cm}^2 \) and the scale factor \( k = \frac{1}{2} \), we can calculate the new surface area as follows:

\[ \text{New Surface Area} = \text{Original Surface Area} \times k^2 \]

Substituting in the values:

\[ \text{New Surface Area} = 54\pi \times \left(\frac{1}{2}\right)^2 \] \[ = 54\pi \times \frac{1}{4} \] \[ = \frac{54\pi}{4} \] \[ = 13.5\pi \]

Thus, the surface area of the new cylinder after dilation is \( 13.5\pi , \text{cm}^2 \).

Answer 2

answers are
6.75
108
27
13.5

Answer 3

Thank you for providing the options. Based on the calculation previously provided, we determine that the surface area after dilation turns out to be \( 13.5\pi , \text{cm}^2\). To determine which option might be correct, we can evaluate the numerical value of \( 13.5\pi \).

Using \( \pi \approx 3.14 \):

\[ 13.5\pi \approx 13.5 \times 3.14 \approx 42.39 , \text{cm}^2 \]

Considering this, it's clear that \( 13.5\pi \) itself is not a match for any of the options you listed (6.75, 108, 27, 13.5) as they are likely linear values, not multiples of \(\pi\).

If we take your options literally and consider \( 13.5 \) without \( \pi \), then \( 13.5 \) is indeed an option listed.

However, the correct scaled surface area in terms of \(\pi\) is \( 13.5\pi \) cm², which corresponds to an approximate value of \( 42.39 \) cm².

If you are looking for the dimensionless value without reference to \(\pi\), then \( 13.5 \) is the correct choice to select from your list.