Sure! To find the radius of the cone given its volume and height, we'll go through the following steps:
1. Understand the Formula for the Volume of a Cone:
The volume [tex]\( V \)[/tex] of a cone can be computed using the formula:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
where:
- [tex]\( V \)[/tex] is the volume,
- [tex]\( \pi \)[/tex] is a constant (approximately 3.14159),
- [tex]\( r \)[/tex] is the radius of the base of the cone,
- [tex]\( h \)[/tex] is the height of the cone.
2. Given Values:
We are given:
[tex]\[
V = 19,200 \, \text{dm}^3
\][/tex]
[tex]\[
h = 16 \, \text{dm}
\][/tex]
3. Rearrange the Formula to Solve for [tex]\( r \)[/tex]:
To find the radius [tex]\( r \)[/tex], we need to rearrange the volume formula. Start by solving for [tex]\( r^2 \)[/tex]:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
Multiply both sides by 3 to get rid of the fraction:
[tex]\[
3V = \pi r^2 h
\][/tex]
Divide both sides by [tex]\( \pi h \)[/tex]:
[tex]\[
r^2 = \frac{3V}{\pi h}
\][/tex]
4. Substitute the Given Values:
Substitute the values of [tex]\( V = 19,200 \, \text{dm}^3 \)[/tex] and [tex]\( h = 16 \, \text{dm} \)[/tex] into the formula:
[tex]\[
r^2 = \frac{3 \cdot 19,200}{\pi \cdot 16}
\][/tex]
5. Calculate [tex]\( r^2 \)[/tex]:
Evaluate the expression:
[tex]\[
r^2 \approx \frac{57,600}{50.2655} \approx 1,145.9155902616465
\][/tex]
6. Find [tex]\( r \)[/tex] by Taking the Square Root:
To find [tex]\( r \)[/tex], take the square root of [tex]\( r^2 \)[/tex]:
[tex]\[
r \approx \sqrt{1,145.9155902616465} \approx 33.851375012865375 \, \text{dm}
\][/tex]
Hence, the radius of the cone is approximately [tex]\( 33.85 \)[/tex] dm.
