Multiple choice

A circular plate expands when heated from a radius of $5 \ cms$ to $5.06 \ cm$ then the approximate percentage increase in its area is?

  1. $0.6$
  2. $1.2$
  3. $2.4$
  4. $0.12$
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C Correct answer
Explanation

The area of a circle is given by A = pi * r^2. Using differentials, the relative change in area is dA/A = 2 * (dr/r). Substituting r = 5 cm and dr = 0.06 cm, we get dA/A = 2 * (0.06 / 5) = 0.024, which corresponds to a 2.4% increase.

AI explanation

The area of a circle is pi multiplied by the radius squared, so we use differentials to find the approximate increase in the area. The derivative of the area is 2 multiplied by pi and the radius, so the change in area equals 2 times pi times 5 multiplied by 0.06, which is 0.6 times pi. Dividing this increase by the original area of pi multiplied by 5 squared gives 0.6 times pi divided by 25 times pi, resulting in a decimal of 0.024. Multiplying by 100 gives a 2.4 percentage increase in the area.

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