ma.912.gr.4.3
volume, surface area, and dilations
use the table below to answer the following questions.
original cylinder
dilation with
scale factor k
new
surface
area
new
volume
radius of the base is 5 inches
height of the cylinder is 12
inches
surface area =______in²
inches volume =______in³
k = 2
k = 3
k = 1/2
part a. determine the surface area and volume of the given
cylinder.
part b. given the three different dilations determine the new
surface areas and volumes.

Question

Accepted Answer

# Explanation:
## Part A
### Surface Area
- **Step1: Recall the formula for the surface area of a cylinder**
The formula for the surface area of a cylinder is $S = 2\pi r(r + h)$, where $r$ is the radius and $h$ is the height.
- **Step2: Substitute the given values**
Given $r = 5$ inches and $h = 12$ inches.
$$
\begin{align*}
S&=2\pi\times5\times(5 + 12)\
&=10\pi\times17\
&=170\pi\
&\approx170\times3.14\
& = 533.8
\end{align*}
$$
### Volume
- **Step1: Recall the formula for the volume of a cylinder**
The formula for the volume of a cylinder is $V=\pi r^{2}h$.
- **Step2: Substitute the given values**
Given $r = 5$ inches and $h = 12$ inches.
$$
\begin{align*}
V&=\pi\times5^{2}\times12\
&=\pi\times25\times12\
&=300\pi\
&\approx300\times3.14\
&=942
\end{align*}
$$

## Part B
### For $k = 2$
- **Surface Area**
The formula for the surface area of a dilated cylinder is $S_{new}=k^{2}S_{original}$.
Since $S_{original}=170\pi\approx533.8$ and $k = 2$, then $S_{new}=2^{2}\times533.8 = 4\times533.8=2135.2$
- **Volume**
The formula for the volume of a dilated cylinder is $V_{new}=k^{3}V_{original}$.
Since $V_{original}=300\pi\approx942$ and $k = 2$, then $V_{new}=2^{3}\times942=8\times942 = 7536$

### For $k = 3$
- **Surface Area**
Using $S_{new}=k^{2}S_{original}$, with $S_{original}=170\pi\approx533.8$ and $k = 3$
$S_{new}=3^{2}\times533.8=9\times533.8 = 4804.2$
- **Volume**
Using $V_{new}=k^{3}V_{original}$, with $V_{original}=300\pi\approx942$ and $k = 3$
$V_{new}=3^{3}\times942=27\times942=25434$

### For $k=\frac{1}{2}$
- **Surface Area**
Using $S_{new}=k^{2}S_{original}$, with $S_{original}=170\pi\approx533.8$ and $k=\frac{1}{2}$
$S_{new}=(\frac{1}{2})^{2}\times533.8=\frac{1}{4}\times533.8 = 133.45$
- **Volume**
Using $V_{new}=k^{3}V_{original}$, with $V_{original}=300\pi\approx942$ and $k=\frac{1}{2}$
$V_{new}=(\frac{1}{2})^{3}\times942=\frac{1}{8}\times942=117.75$

# Answer:
- **Part A**
    - Surface Area: $533.8\space in^{2}$
    - Volume: $942\space in^{3}$
- **Part B**
    - For $k = 2$:
        - New Surface Area: $2135.2\space in^{2}$
        - New Volume: $7536\space in^{3}$
    - For $k = 3$:
        - New Surface Area: $4804.2\space in^{2}$
        - New Volume: $25434\space in^{3}$
    - For $k=\frac{1}{2}$:
        - New Surface Area: $133.45\space in^{2}$
        - New Volume: $117.75\space in^{3}$

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QUESTION IMAGE

Question

Explanation:

Part A

Surface Area

Volume

Part B

For \(k = 2\)

For \(k = 3\)

For \(k=\frac{1}{2}\)

Answer: