Multiple choice

The diameter of a circle found by measurement $5.2$ cms with a maximum error $0.05$ cms. The maximum error in its area is?

  1. $4.1$ sq.cms
  2. $0.041$ sq.cms
  3. $0.41$ sq.cms
  4. $0.5$ sq.cms
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Area A = pi * r^2 = pi * (d/2)^2 = (pi/4) * d^2. The error in area dA = (pi/4) * 2 * d * dd = (pi/2) * d * dd. Substituting d=5.2 and dd=0.05 gives (3.14159/2) * 5.2 * 0.05 = 0.408, which rounds to 0.41.

AI explanation

The area of a circle is pi multiplied by the radius squared, so the approximate error in the area can be found using differentials. Taking the derivative of the area gives 2 multiplied by pi and the radius, meaning the approximate error is 2 times pi times r times the error in the radius. The radius is 5.2 divided by 2, which is 2.6 centimeters, and the radius error is 0.05 divided by 2, which is 0.025 centimeters. Multiplying 2 by 3.14, 2.6 and 0.025 gives a maximum error in the area of approximately 0.4082, which rounds to 0.41 square centimeters.

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