let 'x' and 'y' be the dimensions of the rectangle

lets assume that the house is on the side with length equal to y

2x + y = 80

y = 80 - 2x

the area of the rectangle is A = x*y

plug y = 80 - 2x into A = x*y

A = x(80 - 2x)

A = 80x - 2x^2

we need to find the maximum of the parabolic function: A = 80x - 2x^2

since the quotient in front of x^2 is negative (-2 < 0) the function has maximum equal to:

A max = c - b^2/4a, where a = -2, b = 80, c= 0

A max = 0 - 80^2/(4(-2))

A max = 800 ft^2 is the maximum area that she can enclose

by solving 80x - 2x^2 = 800 we find:

x = 20 ft

click here to see the step by step solution of the equation:

y = 80 - 2*20

y = 40 ft

the dimensions of the garden are 20 ft and 40 ft.