Question 229541
Let x = the radius of the first circle
Let y = the radius of the second circle circle<br>
Since perimeter (better known as circumference) of a circle is {{{2pi*r}}} and area is {{{pi*r^2}}} we get the following equations:
{{{2pi*x + 2pi*y = 12pi}}}
{{{pi*x^2 + pi*y^2 = 20pi}}}<br>
Since the second equation is not linear, we'll use the Substitution Method to solve this system. We'll solve the first equation for y. Start by subtracting {{{2pi*x}}} from each side:
{{{2pi*y = 12pi-2pi*x}}}
Divide both sides by {{{2pi}}}:
{{{y = 6 - x}}}
Now we'll substitute this into the other equation:
{{{pi*x^2 + pi*(6-x)^2 = 20pi}}}
Simplifying. We'll get rid of the {{{pi}}}'s by dividing both sides by it:
{{{x^2 + (6-x)^2 = 20}}}
Since this is a quadratic equation we'll simplify and get one side equal to zero:
{{{x^2 + 36 - 12x + x^2 = 20}}}
{{{2x^2 -12x +36 = 20}}}
{{{2x^2 - 12x +16 = 0}}}
Now we solve this by factoring or by using the quadratic equation. This will factor fairly easily:
{{{2(x^2-6x+8) = 0}}}
{{{2(x-4)(x-2) = 0}}}
By the Zero Product Property we know that this product can be zero only if one of the factors is zero. The factor of 2 cannot be zero but the other two factors can:
{{{x-4 = 0}}} or {{{x-2 = 0}}}
which gives us:
{{{x = 4}}} or {{{x = 2}}}<br>
Remember to answer the question. (With word problems it is always tempting to feel satisfaction with having solved for the variable and forget to answer the question.<br>
The question here is to find the two radii. Since x is the radius of one circle, we need to find y, the radius of the other circle, too. We can get y using our x values and the equation y = 6 - x.
For x = 4:
y = 6 - 4 = 2
For x = 2
y = 6 - 2 = 4<br>
So even though we came up with 2 x values, there is only one solution. The radii are 2 and 4.