Answer:
[tex]\huge\boxed{\text{6 by 10 inches}}[/tex]
Step-by-step explanation:
In order to solve for this, we can make two equations, where a is the length of the first side and b is the length of the second side.
- [tex]ab=60[/tex] (Area)
- [tex]2a+2b=32[/tex] (Perimeter)
We can now solve this systems of equations by using substitution.
Let's change the equation [tex]ab=60[/tex] into the form [tex]a= mb+c[/tex].
We can divide both sides by b.
[tex]a = \frac{60}{b}[/tex]
We can now plug [tex]\frac{60}{b}[/tex] in as a into [tex]2a+2b=32[/tex].
[tex]2(\frac{60}{b}) + 2b = 32[/tex]
Let's simplify this equation:
- [tex]\frac{120}{b} + 2b = 32[/tex]
- [tex]2b^2 + 120 = 32b[/tex]
- [tex]2b^2 - 32b + 120 = 0[/tex]
- [tex]b^2 - 16b + 60 = 0[/tex]
- [tex]-10 \cdot -6 = 60\\-10 + -6 = -16[/tex] (Quadratic factoring, [tex]x_1 \cdot x_2=c, x_1 + x_2 = b[/tex])
- [tex](b-10)(b+6)[/tex]
- [tex]b = 10\ \text{or}\ 6[/tex]
Now, since we have both roots of this quadratic, we know that the two sides will be 10 and 6 inches long.
Hope this helped!
