Final answer:
The radius of the spherical balloon is increasing at a rate of 1 / (25π) cm/s when its diameter is 50 cm, calculated using related rates from calculus.
Step-by-step explanation:
How Fast Is the Radius of the Balloon Increasing?
The student's question involves the rate at which the radius of a spherical balloon increases as air is pumped into it, which is a problem in related rates in calculus. We're given that the volume of the balloon increases at a rate of 100 cm³/s and we need to find the rate at which the radius increases when the diameter is 50 cm. To solve this, we use the formula for the volume of a sphere, V = ⅔πr³, and differentiate both sides with respect to time (t) to establish a relationship between the rates of change of the volume (dV/dt) and the radius (dr/dt).
Given: dV/dt = 100 cm³/s and diameter = 50 cm so radius r = 25 cm
1. Differentiate the volume of a sphere with respect to time: dV/dt = 4πr² (dr/dt)
2. Solve for dr/dt: dr/dt = dV/dt / (4πr²)
3. Substitute the known values: dr/dt = 100 / (4π(25)²)
4. Calculate the rate at which the radius increases: dr/dt = 100 / (4π(625)) cm/s
5. Simplify and get the final answer: dr/dt = 1 / (25π) cm/s
Thus, the radius of the balloon is increasing at a rate of 1 / (25π) cm/s when the diameter is 50 cm.