calculus question

Question

Helium is being pumped into a spherical balloon at a rate of 3 cubic centimeters per second. How fast is the radius of the balloon increasing when the radius is 10 centimeters?

Round your answer to four decimal places (please write final answer).

Answer 1

Given information

Volume of Sphere: V = (4/3)πr³

Rate of volume with respect to time: dV/dt = 3 cm³/s

Radius: r = 10 cm

Rate of radius with respect to time: dr/dt_r=10 = ?

Evaluate

V = (4/3)πr³

dV/dt = 4πr²(dr/dt)

3 = 4π(10)²(dr/dt)

3 = 400π(dr/dt)

3/(400π) = dr/dt

dr/dt ≈ 0.0024

Therefore, the radius of the balloon is increasing at a rate of 3/(400π), or about 0.0024 cubic centimeters per second when the radius is 10 centimeters.

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