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Calculus question please explain answer provided?
A rubber beach ball was being inflated at a rate of 28ft^3 /min, but burst wehn volume reached 256pi/81ft^3. At what rate was the radius expanding when the balloon burst.
Answer Max area 16 sq. units
The answer doesn't match the question. The question asks for a rate of change of radius, which should be in ft/min in this problem, but the answer is an area. Use the relationship between radius and volume:
V = (4/3)πr³
Then take d/dt of both sides:
dV/dt = (4πr²) dr/dt
...but 4πr² = (4/3)πr³ * (3/r) which is V * 3/r, so:
dV/dt = V * 3/r * dr/dr
dr/dt = (r/3) * (dV/dt) / V
Finally, reverse the volume formula to get:
r = ∛[ 3V/(4π) ]
Since V and dV/dt are given, you have the material for an answer:
r = ∛[ 3 * (256π/81)/(4π)] = ∛[64/27] = 4/3 ft.
dr/dt = (r/3) (dV/dt) / V = (4/9) * (28) / (256π/81)
= (112 / 9) * (81 / 256) * 1/π = 63/(16π) ft/min