If the rate of change of radius of a circle is 6 cm/s then find the rate of change of area of the circle when r=2 cm.a)74.36 cm2/sb)75.36 cm2/sc)15.36 cm2/sd)65.36 cm2/sCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

The rate of change of radius of the circle is dr/dt = 6 cm/s
The area of a circle is A=πr²
Differentiating w.r.t t we get,
(dA/dt = d/dt) (πr²) = 2πr (dr/dt) =2πr(6)=12πr.
dA/dt |_r=2=24π= 24×3.14=75.36 cm²/s

Answer

To find the rate of change of the area of a circle, we need to use the formula for the area of a circle, which is A = πr^2, where A is the area and r is the radius.

Given that the rate of change of the radius is 6 cm/s, we can differentiate both sides of the equation with respect to time to find the rate of change of the area:

dA/dt = d/dt (πr^2)

Using the chain rule, we can differentiate the right side of the equation:

dA/dt = 2πr (dr/dt)

Substituting the given value of dr/dt = 6 cm/s and r = 2 cm, we can calculate the rate of change of the area:

dA/dt = 2π(2)(6) = 24π

Now, we can approximate the value of π as 3.14 to find the value of dA/dt:

dA/dt ≈ 24(3.14) ≈ 75.36 cm^2/s

Therefore, the rate of change of the area of the circle when r = 2 cm is approximately 75.36 cm^2/s.

Therefore, option B, 75.36 cm^2/s, is the correct answer.

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