A large spherical weather balloon is being inflated at 0.04
𝑚
3
𝑠

What is the rate of change of the surface area of the balloon with respect to time when the radius of the balloon is 0.9 m?

You should assume SI units and enter your answer to 2 decimal places rounding as appropriate (there is no need to enter the units after your value).

Hint: you will need to look up the formulae for both the surface area of a sphere and for the volume of a sphere.

Question

A large spherical weather balloon is being inflated at  0.04 
𝑚
3
𝑠

What is the rate of change of the surface area of the balloon with respect to time when the radius of the balloon is 0.9 m?

You should assume SI units and enter your answer to 2 decimal places rounding as appropriate (there is no need to enter the units after your value).

Hint: you will need to look up the formulae for both the surface area of a sphere and for the volume of a sphere.

BlackTom AI · Accepted Answer

Surface area A = 4πr^2 and volume V = (4/3)πr^3. Relate rates using the chain rule: dV/dt = (dV/dr)(dr/dt) and dA/dt = (dA/dr)(dr/dt).

1) Differentiate the volume with respect to r:
dV/dr = 4πr^2.
Given dV/dt = 0.04 m^3/s, solve for dr/dt:
dr/dt = (dV/dt) / (dV/dr) = 0.04 / (4πr^2).

Substitute r = 0.9 m:
4πr^2 = 4π(0.9)^2 = 4π(0.81) = 3.24π.
Thus dr/dt = 0.04 / (3.24π) ≈ 0.04 / 10.183 = 0.00392697... m/s.

2) Differentiate the surface area with respect to r:
dA/dr = 8πr.
Then dA/dt = (dA/dr)(dr/dt) = 8πr · dr/dt.

Substitute r = 0.9 and the expression for dr/dt:
dA/dt = 8π(0.9) · [0.04 / (3.24π)].
Cancel π: dA/dt = (8·0.9 / 3.24) · 0.04 = (7.2 / 3.24) · 0.04.

Compute the fraction: 7.2 / 3.24 = 2.222222...
So dA/dt = 2.222222... × 0.04 = 0.0888889... m^2/s.

Rounded to two decimal places: 0.09 (units m^2/s; the answer input should be 0.09).

Why this is correct: the method uses correct geometric formulas and the chain rule to convert the given volumetric inflation rate to the rate of change of surface area. A common error would be differentiating incorrectly (e.g., using wrong formulas or forgetting to use dr/dt) or failing to cancel π and simplify, which would give a different numeric result.

Final confirmation: the computed dA/dt rounded to two decimals is 0.09, which matches the provided answer.

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