To find the volume of a rectangular pyramid, you can use the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
- First, calculate the base area of the rectangular base with dimensions 12 inches by 20 inches:
\[ \text{Base Area} = \text{Length} \times \text{Width} = 12 , \text{in} \times 20 , \text{in} = 240 , \text{in}^2 \]
- Next, plug the base area and the height (31 inches) into the volume formula:
\[ V = \frac{1}{3} \times 240 , \text{in}^2 \times 31 , \text{in} \]
- Now calculate it step by step:
\[ V = \frac{1}{3} \times 240 \times 31 \]
\[ 240 \times 31 = 7440 \]
\[ V = \frac{7440}{3} = 2480 , \text{in}^3 \]
Thus, the volume of the sign is
\[ \boxed{2480 , \text{in}^3} \]