How fast is the balloon's surface area growing…

How fast is the balloon's surface area growing when the radius is 5 cm?

Question 15

MediumOriginal question

Air is pumped into a spherical balloon at a steady 100 cm³ every second. For a sphere of radius $r$ , $V = \frac{4}{3}\pi r^{3}$ and the surface area is $S = 4\pi r^{2}$ .

How fast is the balloon's surface area growing when the radius is 5 cm?

20 cm²/sCorrect answerYour answer40 cm²/sCorrect answerYour answer80 cm²/sCorrect answerYour answer

40 \pi

cm²/sCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

$\frac{dS}{dt} = 8\pi r\frac{dr}{dt}$ .
At $r = 5$ , $\frac{dr}{dt} = \frac{1}{\pi}$ : $8\pi(5)\left(\frac{1}{\pi}\right) = 40$ cm²/s.

Worth noting

$40\pi$ forgets that $\frac{dr}{dt}$ already carries a $\frac{1}{\pi}$ .

Concept: Related rates: area through the radius
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsRate of change of a quantity

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Ten questions from Rate of Change, Marginal Cost & Revenue, at your own pace.

How fast is the balloon's surface area growing when the radius is 5 cm?

Question 15

Explanation