Question

A large cylindrical cooler is 2 (1 )/2 feet high and has a diameter of 1 1/2 feet. It is filled 3/4 high with water for athletes to use during their soccer game. Estimate the volume of the water in the cooler to the nearest tenth of a gallon. (Hint: 1 gallon ≈ 231 in3)

Answer 1

So.. if hmm check the picture below, the radius is half that diameter

thus $\bf \textit{volume of a cylinder}\\\\ V=\pi r^2 h\qquad \begin{cases} h=height\\ r=radius=\frac{diameter}{2}\\ ----------\\ h=2\frac{1}{2}\to \cfrac{5}{2}\\ diameter=1\frac{1}{2}\to \cfrac{3}{2}\\ r=\cfrac{\frac{3}{2}}{2}\to \cfrac{3}{4} \end{cases}\implies V=\pi \left( \cfrac{3}{4} \right)^2\cfrac{5}{2} \\\\\\ V=\pi \cdot \cfrac{9}{16}\cdot \cfrac{5}{2}\impliedby \textit{how much is }\frac{3}{4}\textit{ of that?} \\\\\\ \cfrac{3}{4}\left( \pi \cdot \cfrac{9}{16}\cdot \cfrac{5}{2} \right)$

simplify that away

now.. that's how much is in the cooler in cubic feet though, not in gallons

notice, our units were in feet, thus our volume is also in feet, just cubics

so. to know how many gallons, simply divide that by 231

$\bf \cfrac{\frac{3}{4}\left( \pi \cdot \frac{9}{16}\cdot \frac{5}{2} \right)}{231}\quad gallons$