To find the length of the base edge [tex]\( a \)[/tex] of a right square pyramid that is 8 inches tall as a function of its volume [tex]\( v \)[/tex] in cubic inches, we will use the formula for the volume of a pyramid.
The volume [tex]\( V \)[/tex] of a pyramid can be given by:
[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]
Since the base of the pyramid is a square with side length [tex]\( a \)[/tex], the base area will be:
[tex]\[ \text{Base Area} = a^2 \][/tex]
The height [tex]\( h \)[/tex] of the pyramid is given as 8 inches. Plugging these into the volume formula, we get:
[tex]\[ V = \frac{1}{3} \times a^2 \times 8 \][/tex]
Simplify the equation:
[tex]\[ V = \frac{8}{3} a^2 \][/tex]
We want to solve for [tex]\( a \)[/tex] in terms of [tex]\( v \)[/tex], so we start by isolating [tex]\( a^2 \)[/tex]:
[tex]\[ a^2 = \frac{3 V}{8} \][/tex]
Now, to obtain [tex]\( a \)[/tex], we take the square root of both sides:
[tex]\[ a = \sqrt{\frac{3 V}{8}} \][/tex]
Since [tex]\( V \)[/tex] is mentioned in the problem as [tex]\( v \)[/tex], we can replace [tex]\( V \)[/tex] with [tex]\( v \)[/tex]. Therefore:
[tex]\[ a(v) = \sqrt{\frac{3 v}{8}} \][/tex]
Thus, the correct answer is:
[tex]\[ D. a(v)=\sqrt{\frac{3 v}{8}} \][/tex]
