Multiple choice

A circular metal plate is heated so that its radius increases at a rate of $0.1$ $mm/ minute$. Then the rate at which the plate's area is increasing when the radius is $50$ $cm$ is

  1. $10 \pi$ $mm^2 /minute$
  2. $100 \pi$ $mm^2 /minute$
  3. $ \pi$ $mm^2 /minute$
  4. $- \pi$ $mm^2 /minute$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Area A = pi * r^2. Rate of change dA/dt = 2 * pi * r * (dr/dt). Given r = 50 cm = 500 mm and dr/dt = 0.1 mm/min. dA/dt = 2 * pi * 500 * 0.1 = 100 * pi mm^2/min.

AI explanation

The area of a circular plate is given by the formula A = pi * r squared, so its rate of change is found by differentiating with respect to time: dA/dt = 2 * pi * r * dr/dt. We convert the radius of 50 cm to 500 mm to match the rate units. Substituting r = 500 mm and dr/dt = 0.1 mm/minute gives dA/dt = 2 * pi * 500 * 0.1. The rate at which the plate's area is increasing is therefore 100 * pi mm^2/minute.

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