Quadratic equations word problem.

Question

A uniform walkway is built around a rectangular flower bed that is 20m by 40m. There is enough material to make a walkway that has a total area of 700 m^2. What is the width of the walkway?

• I need help making the quadratic equation, the rest I think I can manage. Thanks to anyone who can help.

Answer 1

There are two rectangles. The inner one with area $A_i$ and outer one with area $A_o$.

$A_i=20\times 40 = 800$

$A_o=(40+2x)(20+2x)=4x^2+120x+800$

And, we know the the total area of the walkway, $A_o-A_i$, is $700$. Combine all this information and we get a quadratic:

$A_o-A_i=4x^2+120x+800-800=4x^2+120x=700$

Solving for $x$:

$4x^2+120x-700=0\implies(x-5)(x+35)=0$ which gives $x=5 $ or $-35$ as the width of the path. $-35$ is a silly answer, so take your path width as $5$.

Excuse the drawing. :)

Answer 2

Or perhaps $2x(20+40)+\pi{x^2}=700$, if your taste runs to rounded corners.