x, y - the lengths of the sides of the pathway
The area of a rectangle is the product of the lengths of its sides. The area is 350 m².
[tex]xy=350[/tex]
The perimeter of a rectangle is two times the sum of the lengths of its sides. The perimeter is 90 m.
[tex]2(x+y)=90 \ \ \ \ \ \ |\div 2 \\
x+y=45 \ \ \ \ \ \ \ \ \ \ |-x \\
y=45-x[/tex]
Substitute 45-x for y in the first equation:
[tex]x(45-x)=350 \\
45x-x^2=350 \\
-x^2+45x-350=0 \\
-x^2+10x+35x-350=0 \\
-x(x-10)+35(x-10)=0 \\
(-x+35)(x-10)=0 \\
-x+35=0 \ \lor \ x-10=0 \\
x=35 \ \lor \ x=10[/tex]
[tex]y=45-x \\
\hbox{for } x=35: \\
y=45-35=10 \\ \\
\hbox{for } x=10: \\
y=45-10=35[/tex]
The dimensions of the pathway are 10 m by 35 m.
The length of the pathway is 35 m.
