Answer :
To find the expression that can be used to determine the height [tex]\( h \)[/tex] of the cone, let's start by considering the formula for the volume of a cone, which is:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
We are given:
- Volume [tex]\( V \)[/tex]: [tex]\( 147 \pi \)[/tex]
- Radius [tex]\( r \)[/tex]: 7 cm
Plug these values into the formula:
[tex]\[ 147 \pi = \frac{1}{3} \pi (7)^2 h \][/tex]
Now, we need to simplify this equation to solve for [tex]\( h \)[/tex].
1. Start by squaring the radius:
[tex]\[ 7^2 = 49 \][/tex]
2. Substitute back into the equation:
[tex]\[ 147 \pi = \frac{1}{3} \pi \times 49 \times h \][/tex]
3. The [tex]\(\pi\)[/tex] on both sides can be cancelled out:
[tex]\[ 147 = \frac{1}{3} \times 49 \times h \][/tex]
4. Multiply both sides by 3 to get rid of the fraction:
[tex]\[ 147 \times 3 = 49 \times h \][/tex]
5. Simplify the left side:
[tex]\[ 441 = 49h \][/tex]
6. Finally, solve for [tex]\( h \)[/tex] by dividing both sides by 49:
[tex]\[ h = \frac{441}{49} \][/tex]
[tex]\[ h = 9 \][/tex]
Therefore, the expression that can be used to find [tex]\( h \)[/tex] is:
[tex]\[ 147 \pi = \frac{1}{3} \pi \left[ 7^2 \right] h \][/tex]
And from here, solving gives us a height [tex]\( h \)[/tex] of 9 cm. So, the correct expression from the options given corresponds to this process is:
[tex]\[ 147 \pi = \frac{1}{3} \pi \left[ 7^2 \right] h \][/tex]
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
We are given:
- Volume [tex]\( V \)[/tex]: [tex]\( 147 \pi \)[/tex]
- Radius [tex]\( r \)[/tex]: 7 cm
Plug these values into the formula:
[tex]\[ 147 \pi = \frac{1}{3} \pi (7)^2 h \][/tex]
Now, we need to simplify this equation to solve for [tex]\( h \)[/tex].
1. Start by squaring the radius:
[tex]\[ 7^2 = 49 \][/tex]
2. Substitute back into the equation:
[tex]\[ 147 \pi = \frac{1}{3} \pi \times 49 \times h \][/tex]
3. The [tex]\(\pi\)[/tex] on both sides can be cancelled out:
[tex]\[ 147 = \frac{1}{3} \times 49 \times h \][/tex]
4. Multiply both sides by 3 to get rid of the fraction:
[tex]\[ 147 \times 3 = 49 \times h \][/tex]
5. Simplify the left side:
[tex]\[ 441 = 49h \][/tex]
6. Finally, solve for [tex]\( h \)[/tex] by dividing both sides by 49:
[tex]\[ h = \frac{441}{49} \][/tex]
[tex]\[ h = 9 \][/tex]
Therefore, the expression that can be used to find [tex]\( h \)[/tex] is:
[tex]\[ 147 \pi = \frac{1}{3} \pi \left[ 7^2 \right] h \][/tex]
And from here, solving gives us a height [tex]\( h \)[/tex] of 9 cm. So, the correct expression from the options given corresponds to this process is:
[tex]\[ 147 \pi = \frac{1}{3} \pi \left[ 7^2 \right] h \][/tex]
