Problema Solution
A farmer wants to enclose a rectangular field with fencing. The fencing along the north side of the field will cost $6 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 $4 per foot, 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 $420.
Answer provided by our tutors
let
l = the length of the rectangular field
w = the width of the rectangular field
north side cost: 6l
south side cost: 6l
east side cost: 4w
west side cost: 10w
6l + 6l + 4w + 10w = 120
2(6l + 7w) = 120 divide both sides by 2
6l + 7w = 60 => l = 10 - (7/6)w
the area of the rectangle A = l*w
A = (10 - (7/6)w)w
A = - (7/6)w^2 + 10w
we need to find the values for w and l so that A has maximum
Since A is a quadratic function with the negative coefficient in front of w^2 follows A has maximum in the vertex equal to
c - b^2/(4a) where a = -7/6, b = 10 and c = 0
A max = - 10^2/(4*(-7/6)) =100*6/28 = 150/7
click here to see the graph of the function
if we solve the equation
- (7/6)w^2 + 10w = 150/7
we find
w = 30/7 ft = 4.29 ft
click here to see the step by step solution of the quadratic equation
l = 10 - (7/6)w
l = 10 - (7/6)*(30/7)
l = 5 ft
the width of the rectangle is 4.29 ft and the length is 5 ft.