Problema Solution
A farmer wants to enclose a rectangular field with fencing. The fencing along the north side of the field will cost $10 per foot; the fencing along the south side of the field will cost $6 per foot; the fencing along the east side of the field will cost $10 per foot; and the fencing along the west side of the field will cost $10 per foot. The farmer wants to enclose the largest area he can without spending more than $150.
2.1
How long should he make the fence along the north side of the field?
Give your answer as a decimal accurate within ±0.2% of the exact answer.
ANSWER:
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2.2
How long should he make the fence along the east side of the field?
Answer provided by our tutors
Let
l = the length of the rectangular field, l>0
w = the width of the rectangular field, w>0
the length on north side cost: 10l
the length on south side cost: 6l
the width on east side cost: 10w
the width on west side cost: 10w
The total cost should not be more than $150 that is:
10l + 6l + 10w + 10w <= 150
16l + 20w <= 150 divide both sides by 2
8l + 10w <= 150
The area of the rectangle A(l, w) = l*w.
We need to find the values for w and l so that A has maximum and also:
l >0
w>0
8l + 10w <= 150
Lets plug l = (1/8)(150 - 10w) into A = l*w and find w so that A has maximum
A = (1/8)(150 - 10w)*w
A = (-5/4)w^2 + 75/4
w = - b/2a, where a = -5/4, b = 75/4
w = -(75/4)/(2*(-5/4))
w = 7.5 ft
l = (1/8)(150 - 10*7.5)
l = 9.375 ft
(9.375, 7.5) satisfy the condition 8l + 10w <= 150 (8*9.375 + 10*7.5 = 150)
The length on north side should be 9.375 ft.
The length on east side should be 7.5 ft.