To find the height of the cylindrical grain bin, we can use the formula for the volume \( V \) of a cylinder:
\[ V = \pi r^2 h \]
where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height.
Given:
- The volume \( V = 31,400 \) cubic feet,
- The diameter of the cylinder \( d = 40 \) feet, which means the radius \( r = \frac{d}{2} = \frac{40}{2} = 20 \) feet.
Now we can substitute π and the values into the volume formula:
\[ 31,400 = 3.14 \times (20^2) \times h \]
Calculating \( 20^2 \):
\[ 20^2 = 400 \]
Now substitute back into the equation:
\[ 31,400 = 3.14 \times 400 \times h \]
Calculating \( 3.14 \times 400 \):
\[ 3.14 \times 400 = 1256 \]
Now the equation is:
\[ 31,400 = 1256h \]
To find \( h \), divide both sides by 1256:
\[ h = \frac{31,400}{1256} \]
Calculating the height \( h \):
\[ h \approx 25 \]
Thus, the height of the grain bin is approximately 25 feet.