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Q. 81

Expert-verified
Found in: Page 774

Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

Two circles have circumferences that add up to $12\mathrm{\pi }\mathrm{cm}$ and areas that add up to $20\mathrm{\pi }{\mathrm{cm}}^{2}$. Find the radius of each circle.

The radius of the two circles are $2and4$.

See the step by step solution

Step 1. Given information.

Assume x to be the radius of the first circle and y to be the radius of the second circle.

From the question,

$2\mathrm{\pi x}+2\mathrm{\pi y}=12\mathrm{\pi }......\left(\mathrm{i}\right)\phantom{\rule{0ex}{0ex}}{\mathrm{\pi x}}^{2}+{\mathrm{\pi y}}^{2}=20\mathrm{\pi }......\left(\mathrm{ii}\right)$

Take out the common factors from equation (i),

$2\mathrm{\pi }\left(\mathrm{x}+\mathrm{y}\right)=2\mathrm{\pi }\left(6\right)\phantom{\rule{0ex}{0ex}}\mathrm{x}+\mathrm{y}=6.......\left(\mathrm{iii}\right)$

Take out the common factors from equation (ii),

$\mathrm{\pi }\left({\mathrm{x}}^{2}+{\mathrm{y}}^{2}\right)=20\mathrm{\pi }\phantom{\rule{0ex}{0ex}}{\mathrm{x}}^{2}+{\mathrm{y}}^{2}=20......\left(\mathrm{iv}\right)$

Step 2. Solve equation (iii) for x.

Solving equation (iii) for x,

$x=6-y$

Substitute the value of x in equation (iv),

localid="1646849857123" role="math" ${\left(6-y\right)}^{2}+{y}^{2}=20\phantom{\rule{0ex}{0ex}}\left(y-4\right)\left(y-2\right)=0\phantom{\rule{0ex}{0ex}}y=4,2$

Substitute $y=4$ in equation (iii),

$x+4=6\phantom{\rule{0ex}{0ex}}x=2$

Substitute $y=2$ in equation (iii),

$x+2=6\phantom{\rule{0ex}{0ex}}x=4$