Let A be the area of a circle with radius r. If dr/dt=3, find dA/dt when r=1. related rates

Expert Answers

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We know that `A=pi r^2,` and in this case the radius and thus the area are functions of `t,` so we use the chain rule and differentiate both sides with respect to `t.` The result is

`(dA)/(dt)=2pi r (dr)/(dt).`

Substituting `r=1` and `(dr)/(dt)=3` in the right side, we get

`(dA)/(dt)=2pi(1)(3)=6pi`` `,

so the area is changing at the rate of `6pi` at the instant ` ``r=1`.

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