15.The area of a circle, A = πr2changes as its radius changes. If the radius changeswith respect to time, the change in area with respect to time is:a) constantb) dAdt= 2πrdrdtc) dAdr= 2πrd) dAdt= 2πr +drdte) impossible to find
Question
15.The area of a circle, A = πr2changes as its radius changes. If the radius changeswith respect to time, the change in area with respect to time is:a) constantb) dAdt= 2πrdrdtc) dAdr= 2πrd) dAdt= 2πr +drdte) impossible to find
Solution
The correct answer is b) dAdt= 2πrdrdt.
Here's why:
The area of a circle is given by the formula A = πr^2. If we want to find how the area changes with respect to time, we need to take the derivative of the area with respect to time.
Using the chain rule of differentiation, we get:
dA/dt = d(πr^2)/dt = 2πr * (dr/dt)
So, the rate of change of the area of the circle with respect to time is 2πr times the rate of change of the radius with respect to time.
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