Answer :
To solve this problem, we need to determine which expression can be used to find the height [tex]\( h \)[/tex] of a cone given its volume and radius.
### Step-by-Step Solution:
1. Understand the Formula:
The volume [tex]\( V \)[/tex] of a cone is given by the formula:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
where [tex]\( r \)[/tex] is the radius of the cone, [tex]\( h \)[/tex] is the height, and [tex]\( \pi \)[/tex] is a constant.
2. Substitute the Given Values:
We are given:
- Volume [tex]\( V = 147 \pi \)[/tex] cubic centimeters
- Radius [tex]\( r = 7 \)[/tex] cm
Substitute these values into the volume formula:
[tex]\[
147 \pi = \frac{1}{3} \pi (7)^2 h
\][/tex]
3. Simplifying the Expression:
To find the expression for height [tex]\( h \)[/tex], we can focus on:
- Squaring the radius: [tex]\( 7^2 = 49 \)[/tex]
- Substitute into the volume equation:
[tex]\[
147 \pi = \frac{1}{3} \pi \times 49 \times h
\][/tex]
4. Set Up to Solve for [tex]\( h \)[/tex]:
The expression simplifies to:
[tex]\[
147 \pi = \frac{1}{3} \pi \times 49 \times h
\][/tex]
5. Identify the Correct Option:
This matches the expression given in one of the options:
[tex]\[
147 \pi = \frac{1}{3} \pi \left(7^2\right) (h)
\][/tex]
Thus, the correct expression from the provided options is:
[tex]\[
147 \pi = \frac{1}{3} \pi (7^2) h
\][/tex]
By following these steps, we determined that this expression can be used to find the height [tex]\( h \)[/tex] of the cone.
### Step-by-Step Solution:
1. Understand the Formula:
The volume [tex]\( V \)[/tex] of a cone is given by the formula:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
where [tex]\( r \)[/tex] is the radius of the cone, [tex]\( h \)[/tex] is the height, and [tex]\( \pi \)[/tex] is a constant.
2. Substitute the Given Values:
We are given:
- Volume [tex]\( V = 147 \pi \)[/tex] cubic centimeters
- Radius [tex]\( r = 7 \)[/tex] cm
Substitute these values into the volume formula:
[tex]\[
147 \pi = \frac{1}{3} \pi (7)^2 h
\][/tex]
3. Simplifying the Expression:
To find the expression for height [tex]\( h \)[/tex], we can focus on:
- Squaring the radius: [tex]\( 7^2 = 49 \)[/tex]
- Substitute into the volume equation:
[tex]\[
147 \pi = \frac{1}{3} \pi \times 49 \times h
\][/tex]
4. Set Up to Solve for [tex]\( h \)[/tex]:
The expression simplifies to:
[tex]\[
147 \pi = \frac{1}{3} \pi \times 49 \times h
\][/tex]
5. Identify the Correct Option:
This matches the expression given in one of the options:
[tex]\[
147 \pi = \frac{1}{3} \pi \left(7^2\right) (h)
\][/tex]
Thus, the correct expression from the provided options is:
[tex]\[
147 \pi = \frac{1}{3} \pi (7^2) h
\][/tex]
By following these steps, we determined that this expression can be used to find the height [tex]\( h \)[/tex] of the cone.
