# If Boy 1 cuts half of a 30m by 40m lawn by cutting strips of equal width around the lawn. How wide should the width of the strips be so that the rectangle left that Boy 2 will have to cut is equal to the amount of lawn Boy 1 cut?

Apr 10, 2016

Width of strip should be 5 m

#### Explanation: Let area of the whole be $A$

Given that inner area - outer area = $\frac{A}{2}$

But $A = 30 \times 40$ so the inner area is such that:

$\frac{30 \times 40}{2} = \left(30 - 2 x\right) \left(40 - 2 x\right)$

$\implies \left(30 \times 40\right) = 2 \left(1200 - 60 x - 80 x + 4 {x}^{2}\right)$

$\implies 1200 = 2400 - 280 x + 8 {x}^{2}$

Divide through out by 8

$150 = 300 - 35 x + {x}^{2}$

${x}^{2} - 35 x + 150 = 0$

$\left(x - 5\right) \left(x - 30\right) = 0$

$x = 30 i s \neg \log i c a l$

$\implies x = 5$

Width of strip should be 5 m
'~~~~~~~~~~~~~~~~~~~~~~~~~~~
Check:

Inner rectangle area $= \left(30 - 2 \left(5\right)\right) \left(40 - 2 \left(5\right)\right)$

$= 20 \times 30 = 600 {m}^{2}$

'..........................................'
Outer ring area

$= \left(40 \times 30\right) - 600 = 600 {m}^{2}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~