Certainly! Let's solve this problem step-by-step.
1. Understand the volumes:
- The volume of a cylinder is calculated using the formula:
[tex]\[
V_{\text{cylinder}} = \pi r^2 h
\][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height of the cylinder.
- The volume of a cone with the same radius and height is calculated using the formula:
[tex]\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\][/tex]
2. Relationship between volumes:
- Notice from the formulas that the volume of a cone is exactly one-third the volume of a cylinder with the same radius and height.
3. Given data:
- We are given that the volume of the cylinder is [tex]\( 1200 \, \text{yd}^3 \)[/tex].
4. Calculate the volume of the cone:
Since the volume of the cone is one-third of the volume of the cylinder:
[tex]\[
V_{\text{cone}} = \frac{1}{3} \times V_{\text{cylinder}}
\][/tex]
5. Substitute the given volume:
[tex]\[
V_{\text{cone}} = \frac{1}{3} \times 1200 \, \text{yd}^3
\][/tex]
6. Perform the arithmetic:
[tex]\[
V_{\text{cone}} = 400 \, \text{yd}^3
\][/tex]
So, the volume of the cone with the same radius and height as the cylinder is [tex]\( 400 \, \text{yd}^3 \)[/tex].
