Question

A spherical balloon is filled with 4500π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is ______.

Options

• `9/7`

• `7/9`

• `2/9`

• `9/2`

Answer 1

A spherical balloon is filled with 4500π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is `underlinebb(2/9)`.

Explanation:

Volume of spherical balloon = V = `4/3 πr^3`

Differentiate both the side, w.r.t 't' we get,

`(dV)/(dt) = 4πr^2 ((dr)/(dt))`  ...(i)

∴ After 49 min,

Volume = (4500 – 49 × 72)π

= (4500 – 3528)π

= 972 π m³

`\implies` V = 972 π m³

∴ 972π = `4/3 πr^3`

`\implies` r³ = 3 × 243 = 3 × 3⁵ = 3⁶ = (3²)³

`\implies` r = 9

Given `(dV)/(dt)` = 72π

Putting `(dV)/(dt)` = 72π and r = 9 in equation (i), we get

∴ 72π = `4π xx 9 xx 9((dr)/(dt))`

`\implies (dr)/(dt) = (2/9)`