Popeye Needs His Spinach

Calculus Level 2

Popeye loves his spinach. He squeezes a can of his favorite spinach in such a way that it retains the shape of a cylinder and its volume remains constant, but the radius of the can decreases at 1 cm 1 \text{ cm} per second.

How fast is the height of the can changing at the moment the can has a radius of 4 cm 4 \text{ cm} and a height of 10 cm ? 10 \text{ cm}?

4 cm/s 5 cm/s 8 cm/s 10 cm/s

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Andrew Ellinor by Andrew Ellinor , 33, USA

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1 solution

We are given that the volume V = π r 2 h V = \pi r^{2}h is constant, so differentiating implicitly with respect to time we find that

d V d t = π ( 2 r d r d t h + r 2 d h d t ) = 0 ⟹ 2 r h d r d t = − r 2 d h d t ⟹ d h d t = − 2 h r d r d t . \dfrac{dV}{dt} = \pi\left(2r\dfrac{dr}{dt}h + r^{2}\dfrac{dh}{dt}\right) = 0 \Longrightarrow 2rh\dfrac{dr}{dt} = -r^{2}\dfrac{dh}{dt} \Longrightarrow \dfrac{dh}{dt} = -\dfrac{2h}{r}\dfrac{dr}{dt}.

So with d r d t = − 1 \dfrac{dr}{dt} = -1 cm/s, we find that when r = 4 c m , h = 10 c m , r = 4 cm, h = 10 cm, we have that

d h d t = − 2 ∗ 10 4 ∗ ( − 1 ) = 5 \dfrac{dh}{dt} = -\dfrac{2*10}{4}*(-1) = \boxed{5} cm/s.

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