Finding the Width using Quadratic equations

Question

The roof needs replacing on a rectangular pavilion that is 8m long and 5m wide. The overhang all the way around the pavilion is to be consistent and the overall roof span cannot be more than 100m2. Find the width of the overhang correct to 2dp.

 

 

Need both length and width equation

Area of the overhang

Equation that equals 0

Showing the HCF

Subbing into quadratic equation ax^2 + bx+ c

+- answer

 

Answer 1

If the overhang is standard all the away around then the

length = 8+x, width = 5+x

 

Area = (8+x)(5+x) ≤ 100

x² + 13x + 40 ≤ 100

x² + 13x - 60 ≤ 0

 

For quadratic x² + 13x - 60 = 0, a=1, b=13, c=-60

 

x = [-13 ±√(13²-4(1)(-60))]/2

x = [-13 ±√(409)]/2

x = (-13±20.224)/2

 

Since x can't be negative disregard (-13-20.224)/2

x = (-13+22.224)/2

x = 3.6118742m

 

Let x = 4

x²+13x-60 = 16+52-60 = 8

Since this is not ≤0

The maximum for the overhang width is 3.61m

 

Note this gives us an area of:

(8+3.61)(5+3.61) = 99.9621 m²