Question 1201846
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Let the dimensions of the rectangle be x and y.  The diagonal of the rectangle is a diameter of the circle, so<br>
[1] {{{xy=150}}}  [the area of the rectangle is 150 sq in]
[2] {{{sqrt(x^2+y^2)=20}}}  [the diameter of the circle is 20 in]<br>
A general problem-solving hint...<br>
Whenever you see {{{xy}}} and {{{x^2+y^2}}} in a problem, look at {{{(x+y)^2=x^2+2xy+y^2}}}.<br>
In this problem....<br>
{{{(x+y)^2=x^2+2xy+y^2=(x^2+y^2)+2(xy)=400+2(150)=700}}}
{{{x+y=sqrt(700)=10*sqrt(7)}}}<br>
Solve that equation for y and substitute in [1]:<br>
{{{y=10*sqrt(7)-x}}}
{{{x(10*sqrt(7)-x)=150}}}
{{{x^2-(10*sqrt(7))x+150=0}}}<br>
That won't factor by usual factoring methods; but the quadratic formula works nicely.<br>
{{{x=(10*sqrt(7)+-sqrt(700-600))/2}}}<br>
{{{x=5*sqrt(7)+5}}} or {{{x=5*sqrt(7)-5}}}<br>
ANSWER: The dimensions of the rectangle are {{{5*sqrt(7)+5}}} and {{{5*sqrt(7)-5}}}<br>
(You can evaluate those expressions and do the prescribed rounding....)<br>