A certain rectangular prism has a height of \(h\) inches, a width that is 2 times its height,
and a length that is 2 inches more than its height. If the area of the base is 48 square inches, what is the volume of the prism, in cubic inches?
Prisms compose of two equal bases that are seperated by height of length \(h\). Below is a sample sketch of a rectangular prism. Note that the base is a rectangle, hence the name rectangular prism.
Since the area of the rectangular base is 48 square inches, the length is 2 more than the height, and the width is twice the height,
$$ A_{\text{base}}= l \cdot w $$
$$ 48= (2+h)(2h) $$
$$ 48=4h+2h^2 $$
$$ 24=2h+h^2$$
$$ h^2+2h-24=0 $$
Solve the quadratic equation using your method of choice. I will factor here:
$$ h^2+2h-24=0 $$
$$ (h+6)(h-4)=0 \tag*{\tiny discard negative heights}$$
$$ h=4 $$
To find the volume of a rectangular prism:
$$ V=A_{\text{base}}\cdot h $$
$$ V= l\cdot w \cdot h $$
$$ V= (2+h)(2h)(h) $$
$$ V = (2+4)(2\cdot 4)(4) $$
$$ V= 6 \cdot 8 \cdot 4 $$
$$ V = \boxed{192} $$
