A toy is formed by placing a solid circular cone on a solid hemisphere and the radius of the common base of the cone and the hemisphere is 15 cm. If the volume of the toy is 2850 $$\pi$$ $$cm^{3}$$, then find the total surface area of that toy.

Solution

If the volume of the toy is 2850 $$\pi$$ $$cm^{3}$$.

volume of the toy = volume of cone + volume of hemisphere

$$2850 \pi = \frac{1}{3}\times\ \pi\ \times\ radius^2\times\ height+\frac{2}{3}\times\ \pi\ \times\ radius^3$$

$$2850\pi=\frac{1}{3}\times\ \pi\ \times\ 15^2\times\ height+\frac{2}{3}\times\ \pi\ \times\ 15^3$$

$$2850=\frac{1}{3}\times\ 225\times\ height+\frac{2}{3}\times3375$$
$$2850=75\times\ height+2\times1125$$
$$2850=75\times\ height+2250$$
$$2850-2250=75\times\ height$$
$$600=75\times\ height$$
height = 8 cm

total surface area of that toy = lateral surface area of cone + curved surface area of hemisphere
= $$\pi\ \times\ radius\times\ \sqrt{radius^2+height^2\ }\ +2\times\ \pi\ \times\ radius^2$$
= $$\pi\ \times\ 15\times\ \sqrt{15^2+8^2\ }\ +2\times\ \pi\ \times\ 15^2$$
= $$\pi\ \times\ 15\times\ \sqrt{225+64\ }\ +2\times\ \pi\ \times225$$
= $$\pi\ \times\ 15\times\ \sqrt{289\ }\ +450\times\ \pi$$
= $$\pi\ \times\ 15\times\ 17\ +450\times\ \pi$$
= $$255\times\ \pi\ +450\times\ \pi$$
= $$705\ \pi$$ $$cm^{2}$$