Example 2 of 7 · Area of a circle

AI SL · Mathematics

A circular pond has radius 20 m, correct to the nearest 10 m. Find the maximum percentage error in the area of the pond.

Smallest true value

15 225π
\[\begin{aligned}T_2&=\pi\times15^2=225\pi\\[6pt]\class{mjx-pe-s6}{\%\text{ error}}&\class{mjx-pe-s6}{{}=\left|\frac{400\pi-225\pi}{225\pi}\right|\times100}\\[6pt]&\class{mjx-pe-s6}{{}=\class{mjx-pe-key}{\boldsymbol{77.8\%}}}\end{aligned}\]

Approximate value

20 400π
\[\begin{aligned}A&=\pi\times20^2=400\pi\\[6pt]\class{mjx-pe-s5}{\%\text{ error}}&\class{mjx-pe-s5}{{}=\left|\frac{A-T}{T}\right|\times100}\end{aligned}\]

Largest true value

25 625π
\[\begin{aligned}T_1&=\pi\times25^2=625\pi\\[6pt]\class{mjx-pe-s7}{\%\text{ error}}&\class{mjx-pe-s7}{{}=\left|\frac{400\pi-625\pi}{625\pi}\right|\times100}\\[6pt]&\class{mjx-pe-s7}{{}=\class{mjx-pe-key}{\boldsymbol{36.0\%}}}\end{aligned}\]

Maximum percentage error = 77.8%. The smallest true value gives the bigger error. The π cancels in each fraction.