Science:Math Exam Resources/Courses/MATH110/December 2014/Question 09 (b)

MATH110 December 2014

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Question 09 (b)

When a pebble is thrown into a pond, circular ripples form on the water surface. Suppose ${\textstyle r(t)}$ is the radius, in centimetres, of one such ripple at ${\textstyle t}$ minutes after the pebble falls into the water, and ${\textstyle A(r)}$ is the surface area of a ripple of radius ${\textstyle r}$. (b) Suppose the radius of the ripple increases at a rate of 20 centimetres per minute. How fast is the area of the ripple increasing when the radius is 10 cm? Include units. (Recall the area of a circle of radius ${\textstyle r}$ is ${\textstyle A=\pi r^{2}}$.)

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Hint
Use the chain rule.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Since the radius of the ripple increases at a rate of 20 centimetres per minute, we have ${\frac {dr}{dt}}=20(cm/min)$. To get ${\frac {dA}{dt}}{\bigg |}_{r=10}$, we use the chain rule first; ${\frac {dA}{dt}}={\frac {dA}{dr}}\cdot {\frac {dr}{dt}}={\frac {d}{dr}}(\pi r^{2})\cdot 20=40\pi r$. Then, plugging $r=10$, we get ${\frac {dA}{dt}}{\bigg |}_{r=10}=40\pi \cdot 10=\color {blue}400\pi (cm^{2}/min)$