Multiple choice

The diameter of a circle is increasing at the rate of $1$ cm/sec. When its radius is $\pi$ , the rate of increase of its area is

  1. $ \pi cm^2 /sec$
  2. $ 2\pi cm^2 /sec$
  3. $ {\pi}^{2} cm^2 /sec$
  4. $ 2{\pi}^{2} cm^2 /sec$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Area A = pi * r^2. Diameter D = 2r, so dD/dt = 2(dr/dt). Given dD/dt = 1, dr/dt = 0.5. dA/dt = 2 * pi * r * (dr/dt). When r = pi, dA/dt = 2 * pi * pi * 0.5 = pi^2.

AI explanation

Using related rates, the area of a circle is A = pi*r^2, so its rate of change is dA/dt = 2*pi*r*(dr/dt). The radius increases at half the rate of the diameter, meaning dr/dt = 0.5 cm/sec. Substituting r = pi and dr/dt = 0.5 into the derivative formula gives dA/dt = 2*pi*(pi)*(0.5), which simplifies to pi^2 cm^2/sec. The result is pi^2 cm^2/sec.

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