The bottom portion of a loading bin is cone shaped. The base radius of this part of the bin is 3.5 feet and the slant height is

Question

2 years ago

15

The bottom portion of a loading bin is cone shaped. The base radius of this part of the bin is 3.5 feet and the slant height is

6.5 feet. What is the capacity and lateral surface area of this part of the bin? Round your answer to the nearest hundredth. Lateral Surface Area = sq. ft. Volume = cu. ft.

Mathematics

2 answers:

jek_recluse [69]2 years ago

7 0

$\bf \textit{lateral surface of a cone}\\\\ LA=\pi r\sqrt{r^2+h^2}~~ \begin{cases} ~~ r=radius\\ sh=\stackrel{slant~height}{\sqrt{r^2+h^2}}\\[-0.5em] \hrulefill\\ r=3.5\\ sh=6.5 \end{cases}\\\\\\ LA=\pi (3.5)(6.5)\implies LA\approx71.47 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{sh}{6.5}=\sqrt{r^2+h^2}\implies 6.5=\sqrt{3.5^2+h^2}\implies 6.5^2=3.5^2+h^2 \\\\\\ 6.5^2-3.5^2=h^2\implies \sqrt{6.5^2-3.5^2}=h\implies \sqrt{30}=h \\\\[-0.35em] ~\dotfill$

$\bf \textit{volume of a cone}\\\\ V=\cfrac{\pi r^2h}{3}~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ r=3.5\\ h=\sqrt{30} \end{cases}\implies V=\cfrac{\pi (3.5)^2\sqrt{30}}{3}\implies V\approx 70.26$