Question

A playground, which measures 50m by 35m, is to be doubled in area by adding a strip of uniform width around the outside of the existing area. What is the width of the new strip around the playground?

Answer 1

`x=(-85+5sqrt(569))/4~~8.567` metres to 3 decimal places

Explanation:

With problems of this type it becomes much clearer if you do a quick sketch.

The inner area is `35 mxx50 m=1750 m^3`

The outer strip area is
`4x^2+(2xx35x)+(2xx50x)=4x^2+170x`

Combining the outer strip and the inner area doubles the inner area. So our model becomes:

`4x^2+170x+1750 = (2xx1750)`

`4x^2+170x+1750 = 3500`

Subtract 3500 from both sides

`4x^2+170x-1750 = 0`

Simplifying the numbers a bit. 4 will not divide exactly into 1750. However, all the numbers are even so we can at least halve them

`2x^2+85x-875=0`

If you can not spot the whole number factors quickly don't waste time in an exam continuing to try and determine them. Use the formula.

`x=(-b+-sqrt(b^2-4ac))/(2a)color(white)("dd")->color(white)("dd") (-85+-sqrt(85^2-4(2)(-875)))/(2(2))`

`x= (-85+-sqrt(14225 ))/4`

`x= (-85+-sqrt(5^2xx569 ))/4 larr" Note that 569 is a prime number"`

`x=(-85+-5sqrt(569))/4 larr" Exact values"`

The negative final value is not logical so is dismissed.

`x=(-85+5sqrt(569))/4 larr" Exact solution"`

`x~~ 8.567151....`

`x=(-85+5sqrt(569))/4~~8.567` to 3 decimal places