Question

I need some help here......100pts......write a quadratic equation for this problem and post a proper answer.

Bryan has 180 feet of fencing to fence off his rectangular garden. He will use the wall of his house as one side of the garden. Write an equation to find the area of the garden. What dimensions create the garden with the greatest area? What is the greatest area?

Answer 1

Explanation:

wall as one side so only 3 sides of fence

let x n y be width n length of garden

length of garden fence = y + 2x

area of garden = x*y

given y + 2x = 180

y = 180 - 2x

substitute into area

area = (180 - 2x)*x = 180x - 2x*x

= 180x - 2x^2

tis the ans 4 quadratic equation

to find max area, use differentiation

d(180x - 2x^2)/dx = 0

180 - 4x = 0

x = 45

y = 180 - 2(45)

= 90

greatest area = 45*90 = 4050 sq ft

Answer 2

let x be the two sides 90deg to the house

house side = 180-2x

Draw a rectangle with length of 2 sides at right angle to house=x

Length of side parallel to house=180-2x

Garden area=x(180-2x)

=180x-2x^2

=-2x^2+180 x

complete the square:

=-2(x^2-90x)

=-2(x^2-90x+2025)+4050

=-2(x-45)^2+4050

so area is max at x-45=0

x=45

House side=180-2*45=90

Garden with the greatest area: 45 by 90 ft

Greatest area=45*90=4050 sq ft