Question 261323
A fence is to be built against a wall. If there are 250 metres of fence available, what are the dimensions that give the maximum area? (shape of fence is a trapezoid) 
height is x
Longer side known as b is x+2y
shorter side known as a is y 
I know that area of a trapezoid is: A=1/2h(a+b) 
please help me! thank you!

a = y
b = x + 2y = x + y + y = x + a + a = 2a + x
b - x = 2a
(b - x)/2 = a
b - 2a = x
b^2 - 4ab + 4a^2 = x^2
A = 1/2 * h * (a + b) = 1/2 * x * (a + 2a + x)
A = 1/2 * x * (3a + x)
A = 1/2 * (x^2 + 3*a*x)
A = 1/2 * x^2 + 3/2 * a * x
A = 1/2 * (b^2 - 4ab + 4a^2) + 3/2 * a * (b - 2a)
A = 1/2 * (b - 2a) * (b - 2a + 3a)
A = 1/2 * (b - 2a) * (a + b)
P = x + a + a + (a + x) + sqrt(x^2 + (a+x)^2)
P = 2x + 3a + sqrt(x^2 + a^2 + 2ax + x^2) = 250
P = 2x + 3a + sqrt(2x^2 + 2ax + a^2) = 250
P = 2(b-2a) + 3a + sqrt(2*(b^2 - 4ab + 4a^2) + 2a(b-2a) + a^2) = 250
P = 2b - 4a + 3a + sqrt(2b^2 - 8ab + 8a^2 + 2ab -4a^2 + a^2) = 250
P = 2b - a + sqrt(2b^2 - 6ab + 5a^2) = 250
P = 2b - a + sqrt(b^2 + (b-a)(b-5a)) = 250
a	b	x	area	perimeter
1	2	0	0	4
2	5	1	3.5	11.16227766
...
34	101	33	2227.5	242.6860094
35	104	34	2363	249.9220385
36	107	35	2502.5	257.1580697
37	110	36	2646	264.394103
did this table in Excel using the formulas found
closest to 250 perimeter is a=35 and b=104 and x=34
so short side a would be 35
long side b would be 104
and height is 34
A = 1/2 * (b - 2a) * (a + b)
A = 1/2 * (104 - 2*35) * (35+104)
A = 1/2 * (104 - 70) * 139
A = 1/2 * 34 * 139
A = 17 * 139
A = 2363 sq.m
P = 2b - a + sqrt(b^2 + (b-a)(b-5a))
P = 2*104 - 35 + sqrt(104^2 + (104-35)*(104-5*35))
P = 208 - 35 + sqrt(10816 + 69*(104-175))
P = 173 + sqrt(10816 + 69*(-71))
P = 173 + sqrt(10816 - 4899)
P = 173 + sqrt(5917)
P = 249.922038 approx 250 m