Question

A footpath of uniform width runs around the outside of a rectangular field 20 m long and 10 m wide. If the path occupies $136m^2,$ its width is .

• A. 2 m
• B. 3 m
• C. 5 m
• D. 17 m

Answer 1

The correct option is A 2 m


$\text{Area of the rectangular field}=20\times 10$
$=200{\text{m}}^2$
Let the width of the footpath be $x$ m.
$\Rightarrow \text{The area of outer rectangle}=(20+2x)(10+2x)$
$=200+40x+20x+4x^2$
$=4x^2+60x+200{\text{m}}^2$

Since the area of the footpath equals the difference between areas of the outer rectangle and the field, we have

$136=4x^2+60x+200-200$

$\Rightarrow x^2+15x-34=0$

i.e., $x^2+17x-2x-34=0$

$\Rightarrow x(x+17)-2(x+17)=0$

$\Rightarrow (x+17)(x-2)=0$

Thus, the roots of the equation are 2 and -17. Since width cannot be negative, it equals 2 m.