How do I solve the following math problem?
You have an 24 cm long string. Examine if you can cut in two parts and create
a) Two squares b) Two circles
whose total area is 20 cm². (The entire length must be used)
It says the string is cut into 2 parts, and not 2 equal parts.
So for 2 squares: The sum of perimeters will be 24 cm. That's,
$4l_1+4l_2=24$ and
$l_1^2+l_2^2=20$
Similarly, For 2 circles:
$2\pi r_1+2\pi r_2=24$
and $\pi r_1^2+\pi r_2^2=20$
I get 2 equations and 2 unknowns, how do I solve these equations?
Thanks!
Note that from the first equation (part a) we have $l_2 = 6-l_1$. Now plug the $l_2$ into the other equation: $l_1^2 + (6-l_1)^2 = 20$. This is a quadratic which we solve: $$l_1^2+(36-12l_1+l_1^2)=20 \\ \Rightarrow 2l_1^2 - 12l_1 + 16 = 0 \\ \Rightarrow l_1^2 - 6l_1 + 8 = 0 \\ \Rightarrow (l_1 - 4)(l_1-2) = 0$$ Therefore $l_1 = 2$ and $l_2=6-l_1=4$ or $l_1=4$ and $l_2=6-l_1=2$. That just means one square has side length 2 and the other has side length 4.
You can apply the same method (called substition) for the second set of equations.