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The diagram below shoes four circles. Let A_1, A_2, A_3, A_4 denote the areas of the red region, yellow region, green region, and blue region, respectively. These areas satisfy A_1 = A_2/2 = A_3/3 = A_4/4. Let r_1 denote the smallest radius and r_4 denote the largest radius. Find r_4/r_1.

Note: A_4 is not the area of the largest circle. It is the area that is inside the largest circle and outside the next-largest circle. Jan 25, 2020

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These areas satisfy A_1 = A_2/2 = A_3/3 = A_4/4.

this can be stated as

$$A_2=2A_1\\ A_3=3A_1\\ A_4=4A_1\\ \text{So the area of the big circle is }\\ Area_{big}=A_1+2A_1+3A_1+4A_1\\ Area_{big}=10A_1\\ \text{So the ratio of the areas of the smallest to biggest circle is }1:10\\ and\\ \text{So the ratio of the radii of the smallest to biggest circles is }1:\sqrt{10}\\$$

So

$$\frac{r_4}{r_1 }=\sqrt{10}$$

You should check this answer. All answers should be checked before they are accepted.

Coding:

A_2=2A_1\\
A_3=3A_1\\
A_4=4A_1\\
\text{So the area of the big circle is }\\
Area_{big}=A_1+2A_1+3A_1+4A_1\\
Area_{big}=10A_1\\
\text{So the ratio of the areas of the smallest to biggest circle is }1:10\\
and\\
\text{So the ratio of the radii of the smallest to biggest circles is }1:\sqrt{10}\\

\frac{r_4}{r_1 }=\sqrt{10}

Jan 26, 2020