Answer
a) $60,501,066.6 \text{ ft}^3$
b) rectangular prism, $114,307,200 \text{ ft}^3$
c) $114,307,200 \text{ ft}^3$, they match
Work Step by Step
a)
$h=200$, $b=756$, $a=314$
$V = \frac{1}{3} \cdot h \cdot (a^2+ab+b^2)$
$V = \frac{1}{3}\cdot 200 \cdot (756^2+756\cdot 314+314^2)$
$V = \frac{200}{3}\cdot (571536+237384+98596)$
$V = \frac{200}{3}\cdot 907516$
$V = 60501066.667$
b)
If $a=b$, then we have a rectangular prism. The shape has the same width and length, and since the height is not the same, we have a rectangular prism.
$V = \frac{1}{3}\cdot h\cdot (a^2+ab+b^2)$
$a=b$
$V = \frac{1}{3}\cdot h\cdot (b^2+b\cdot b+b^2)$
$V = \frac{1}{3}\cdot h\cdot (b^2+b^2+b^2)$
$V = \frac{1}{3}\cdot h\cdot 3\cdot b^2$
$V = 1/3\cdot 3\cdot h\cdot b^2$
$V = b^2\cdot h$
$b=756$, $h=200$
$V = b^2\cdot h$
$V = 756^2\cdot 200$
$V = 571536\cdot 200$
$V = 114307200$
c)
$a=b$, $b=756$, $h=200$
$V = b^2\cdot h$
$V = 756^2\cdot 200$
$V = 571536\cdot 200$
$V = 114307200$
