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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!
Answer by CharlesG2(834)   (Show Source):
You can put this solution on YOUR website! 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
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