ID: Class: 12Subject: MathTopic: Application of DerivativeType: Short
Question:
The radius of a circle is increasing uniformly at the rate of \(3\;{\text{cm}}/{\text{s}}\). Find the rate at which the area of the circle is increasing when the radius is \(10\;{\text{cm}}/{\text{s}}\)
Official Solution
Explanation:
We know that \(A = \pi {r^2}\) \[\therefore \dfrac{{dA}}{{dt}} = \dfrac{d}{{dr}}\left( {\pi {r^2}} \right)\dfrac{{dr}}{{dt}} = 2\pi r\dfrac{{dr}}{{dt}}\] Given that \(\dfrac{{dr}}{{dt}} = 3\;{\text{cm}}/{\text{s}}\) \[\therefore \dfrac{{dA}}{{dt}} = 2\pi r(3) = 6\pi r\] So, when \(r = 10\;{\text{cm}}\), \(\dfrac{{dA}}{{dt}} = 6\pi (10) = 60\pi {\text{c}}{{\text{m}}^2}/{\text{s}}\)
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