Example 3 of 7 · Volume of a cube

AI HL · Mathematics

A cube-shaped storage box has edge length 30 cm, correct to the nearest 10 cm. Find the maximum percentage error in the volume of the box.

Smallest true value

15 625 25
\[\begin{aligned}T_2&=25^3=15\,625\\[6pt]\class{mjx-pe-s6}{\%\text{ error}}&\class{mjx-pe-s6}{{}=\left|\frac{27\,000-15\,625}{15\,625}\right|\times100}\\[6pt]&\class{mjx-pe-s6}{{}=\class{mjx-pe-key}{\boldsymbol{72.8\%}}}\end{aligned}\]

Approximate value

27 000 30
\[\begin{aligned}A&=30^3=27\,000\\[6pt]\class{mjx-pe-s5}{\%\text{ error}}&\class{mjx-pe-s5}{{}=\left|\frac{A-T}{T}\right|\times100}\end{aligned}\]

Largest true value

42 875 35
\[\begin{aligned}T_1&=35^3=42\,875\\[6pt]\class{mjx-pe-s7}{\%\text{ error}}&\class{mjx-pe-s7}{{}=\left|\frac{27\,000-42\,875}{42\,875}\right|\times100}\\[6pt]&\class{mjx-pe-s7}{{}=\class{mjx-pe-key}{\boldsymbol{37.0\%}}}\end{aligned}\]

Maximum percentage error = 72.8%. The smallest true value gives the bigger error, because we divide by it.