To find the height of the cone given its volume, we can use the formula for the volume of a cone:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- \( V \) is the volume of the cone,
- \( r \) is the radius of the base,
- \( h \) is the height of the cone.
We are provided:
- \( V = 3183.96 , \text{cubic yards} \)
- \( \pi \approx 3.14 \)
- The radius \( r = 13 , \text{yd} \)
First, we can calculate \( r^2 \):
\[ r^2 = 13^2 = 169 , \text{yd}^2 \]
Now, substitute \( V \), \( \pi \), and \( r^2 \) into the volume formula:
\[ 3183.96 = \frac{1}{3} \times 3.14 \times 169 \times h \]
To isolate \( h \), first calculate \( \frac{1}{3} \times 3.14 \times 169 \):
\[ \frac{1}{3} \times 3.14 \times 169 \approx 178.198 \]
Now we can rewrite the equation:
\[ 3183.96 = 178.198 \times h \]
Next, solve for \( h \):
\[ h = \frac{3183.96}{178.198} \]
Calculating that:
\[ h \approx 17.87 , \text{yd} \]
Thus, the height of the cone is approximately:
\[ h \approx 17.87 , \text{yards} \]
Rounded to the nearest hundredth, the height \( h \approx 17.87 \) yards.