Question

Find the height in inches of the free surface if 0.8 ft³ of water is poured into a conical tank 20 inches high with a base radius of 10 inches. How much additional water in in³ is required to fill the tank?

_______ inches above the tank Answer in 2 decimal places

_______ in³

Answer 1

Volume of water (V_w) = 0.8 ft³ = 0.8 $\times$ 1728 in³ = 1382.4 in³

Radius of cone (r) = 10 in

height of cone (h) = 20 in

Volume of larger cone

$V_l = \frac{1}{3} \pi r^2 h \\ = \frac{\pi \times 10^2 \times 20}{3} \\ = 2094.39 \;in^3$

Volume of smaller cone = $V_l - V_w$

$= 2094.39-1382.4 \\ = 711.99\; in^3$

from similar triangle concept

$\frac{r_0}{h_0}= \frac{10}{20} \\ r_0=2h_0$

Volume of smaller cone can be written as

$\frac{1}{3} \pi r^2_0h_0 = 711.99 \; in^3$

put the value of r o in the above equation

$\frac{1}{3} \pi (2h_0)^2h_0 = 711.99 \;in^3 \\ h_0 = 5.539 \;in$

thus the height of the free surface = $h- h_0$

$=20-5.539 \\ = 14.461 \;in$

and the additional water required to fill the cone will be equal to the volume of the smaller cone

$V_{addition} = 711.99 \;in^3$