Question

A toy is in the form of a hemisphere surmounting a cone whose radius is $3.5$ cm. If the total height of the toy is $15.5$ cm, find its total surface area and volume.

Hint:

For combined solids, add exposed curved areas and include only the exposed base(s). For volumes, simply sum the volumes of each solid.

Answer 1

Radius $r=3.5$ cm. Total height $=15.5$ cm $\Rightarrow$ cone height $h=15.5-r=12$ cm.
Slant height of cone: $l=\sqrt{r^{2}+h^{2}}=\sqrt{3.5^{2}+12^{2}}=\sqrt{156.25}=12.5$ cm.
Total Surface Area (TSA):
TSA $=$ curved surface of cone $+$ base of cone $+$ curved surface of hemisphere
$\displaystyle =\pi r l+\pi r^{2}+2\pi r^{2}=\pi r l+3\pi r^{2}$.
With $r=3.5$, $l=12.5$ and $\pi=\dfrac{22}{7}$,
$\displaystyle \text{TSA}=\frac{22}{7}\cdot3.5\cdot12.5+3\cdot\frac{22}{7}\cdot(3.5)^{2} =137.5+115.5=253\ \text{cm}^{2}.$
[1mm] \[ \boxed{\text{TSA}=253\ \text{cm}^{2}} \] Volume:
$V=\underbrace{\tfrac{1}{3}\pi r^{2}h}_{\text{cone}}+\underbrace{\tfrac{2}{3}\pi r^{3}}_{\text{hemisphere}} =\tfrac{1}{3}\pi(3.5)^{2}(12)+\tfrac{2}{3}\pi(3.5)^{3} =49\pi+\frac{343}{12}\pi =\frac{931}{12}\pi\ \text{cm}^{3}.
$ With $\pi=\dfrac{22}{7}$, $\displaystyle V=\frac{10241}{42}\approx 243.83\ \text{cm}^{3}.$
[1mm] \[ \boxed{\text{Volume}=\frac{931\pi}{12}\ \text{cm}^{3}\ \ (\approx 243.83\ \text{cm}^{3})} \]