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Problem 448

Algebra Difficulty 2.7 Multiple choice CEMC Gauss (Grade 7) · Canada · 2025

Three circles have radii $1$
cm, $5$ cm, and $x$ cm. If the mean (average) area of the
three circles is $\$30\pi \text{}$
cm}^2 $, the value of$ x$
is

Pick one

Official solution

A circle with radius $1$ cm
has area $\pi\times(1 \text{ cm})^2=\pi \text{ cm}^{2}$ .

A circle with radius $5$ cm has area
$\pi\times(5 \text{ cm})^2=25\pi\text{ cm}^{2}$ .

A circle with radius $x$ cm has area
$\pi\times (x \text{ cm})^2=x^2\pi\text{ cm}^{2}$ .

Since the mean area of the three circles is $30\pi\text{ cm}^{2}$ , then the sum of the areas of the
three circles is $\$3\times30\pi \text{ cm}^2=90\pi\text{ cm}^{2}\$.$

Since $\pi+25\pi+x^2\pi=26\pi+x^2\pi$ , we get
$26\pi+x^2\pi=90\pi$ or $x^2\pi=64\pi$ or $x^2=64$ , and so $x=8$ (since $x>0$ ).

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.