Question

Find the length of the base of a square pyramid if the volume is 576 cubic inches and has a height of 3 inches.

Answer 1

Volume of a square-base pyramid: \(\frac{1}{3}\cdot bh\)
where \(b\) is the area of the Base (\(l\times w\)) or length times width;
and \(h\) is the height (3 inches, as given).

Answer 2

**Forgot to put the volume in the original equation.
So the volume formula is actually \(V=\frac{1}{3}bh\)

Moving on...
As the base is a square, it is equilateral and all lengths/widths are the same. Therefore, we can also rewrite the formula with length as variable \(x\):
where \(b=l\times w\to x\cdot x\to x^{2}\)\[V=\frac{1}{3}\cdot\color{red}{b}h\]thus becomes \(V=\frac{1}{3}\cdot\color{red}{x^{2}}h\), and when we input the \(\color{fuchsia}{\text{volume}}\) and \(\color{lime}{\text{height}}\) value we have\[\color{fuchsia}{576}=\frac{1}{3}\cdot x^{2}(\color{lime}{3})\]

Answer 3

Now we solve for variable \(x\). Let's first see what we can cancel out.
We have a 3 in the numerator (anything without a fraction form is considered numerator) and a 3 in the denominator (the bottom part of the fraction). These cancel out:\[576=\frac{1}{\cancel{3}}\cdot x^{2}(\cancel{3})\]\[576=x^{2}\]

Answer 4

Now we need to isolate the variable to find its value. To do this, we will remove the square by square root.\[\sqrt{576}=\sqrt{x^{2}}\]\[24=x\]So your pyramid base has a length/width of 24 inches.

Why inches? Because volume is in cubic units, and the value given for your volume is in cubic \(\text{inches}\).