Question

determine the volume of the shaded region. when appropriate, use the π key on your calculator. the volume of the shaded region is □ ft³.

Explanation:

Step1: Calculate the volume of the whole rectangular - prism

The volume formula for a rectangular - prism is $V = l\times w\times h$. The dimensions of the whole rectangular - prism are $l = 13$ ft, $w=(7 + 7)=14$ ft, and $h = 8$ ft. So $V_{total}=13\times14\times8=1456$ $ft^{3}$.

Step2: Calculate the volume of the non - shaded triangular - prism

The base of the non - shaded triangular - prism is a right - triangle with base $b = 7$ ft and height $h_{1}=4$ ft. The length of the non - shaded triangular - prism is $l = 13$ ft. The volume formula for a triangular prism is $V=\text{Base Area}\times l$. The base area of the right - triangle is $A=\frac{1}{2}\times7\times4 = 14$ $ft^{2}$. So the volume of one non - shaded triangular prism is $V_{1}=14\times13 = 182$ $ft^{3}$. Since there are 2 non - shaded triangular prisms, $V_{non - shaded}=2\times182 = 364$ $ft^{3}$.

Step3: Calculate the volume of the shaded region

The volume of the shaded region $V_{shaded}=V_{total}-V_{non - shaded}$. So $V_{shaded}=1456 - 364=1092$ $ft^{3}$.

Answer:

$1092$ $ft^{3}$