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Calculus Level 4

If the area of a circle increases at a uniform rate, then the perimeter varies inversely as the radius.

True Cannot Determine Question is wrong False

Shashank Rustagi by Shashank Rustagi , 24, India

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1 solution

Tom Engelsman
Jan 9, 2018

If Area = π r 2 \pi r^2 , then A = π r 2 = r 2 ⋅ 2 π r = 1 2 P r A = \pi r^2 = \frac{r}{2} \cdot 2\pi r = \frac{1}{2} Pr . Solving for the perimeter gives P = 2 A r P = \frac{2A}{r} , which varies inversely as the radius.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Just attempted this one. It is given that area increases at a uniform rate. Then:

A = π R 2 A = \pi R^2 d A d t = 2 π R d R d t \frac{dA}{dt} = 2 \pi R\frac{dR}{dt} ⟹ C = 2 π R d R d t \implies C = 2 \pi R\frac{dR}{dt}

So, this is a governing differential equation to solve for R R as a function of time. Despite solving this, and re-arranging terms to compute perimeter as a function of radius, I still compute P = 2 π R P = 2 \pi R . Hence the answer is false. In your solution, you have not used the information that the area increases at a constant rate. Your solution says that for a given circle area, the perimeter is inversely proportional to the radius.

Karan Chatrath - 3 months, 1 week ago

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