Question

Water flows at the rate of 10 m/minute through a cylindrical pipe 5 mm in diameter.How long would it take to fill a conical vessel whose diamter at the base is 40cm and depth 24cm?

Answer 1

Let the radius of cylindrical pipe be r.

Suppose the radius and height of the conical vessel be R and H respectively.

$\therefore$ r = $\frac{1}{4}$ cm

2R = 40 cm

∴ R = 20 cm

and H = 24 cm

Volume of conical vessel = $\frac{1}{3}$$\pi$$R^2$ H = $\frac{1}{3}$ ×$\pi$ × ${20}^2$ × 24 $c{m}^3$

Rate of water flow = 10 m/min = 1000 cm/min

∴ Volume of water flowing out of the pipe in one minute = $\pi$ $r^2$ × 1000 cm = $\pi$ × ($\frac{1}{4}{)}^2$ × 1000 $c{m}^3$

Let the time taken to fill the conical vessel be t minutes.

∴ Volume of water flowing out of the pipe in tminutes = $\pi$ × ($\frac{1}{4}{)}^2$ × 1000 $c{m}^3$

Volume of water flowing out of the pipe in tminutes = Volume of conical vessel

$\pi$ $\frac{1}{16}$ × 1000 t = $\frac{1}{3}$ $\pi$ 400 × 24

t = $\frac{400\times 24\times 16}{3\times 100}$ = 51.2

Thus, the time taken to fill the conical vessel is 51.2 minutes.