A farmer builds a rectangular grid of pens with 1 row and 5 columns using 850 feet of fencing. What dimensions will maximize the total area of the pen?

Answer 1

2 sides of one Length (1+1)

6 sides of another side = width (5+1) sides

2x+6y = 850

x+3y = 425

x = 425 -3y

Area = A= xy= (425-3y)y = 425y - 3y^2

take the derivative and set equal to zero

A'= 425 -6y = 0

solve for y

6y = 425

y = 425/6 = 70 5/6 feet wide

x = 425 -3(425/6) = 212 1/2 feet Long

Length = 3 times width. 3 times (425/6) = 425/2 = 212 1/2 feet

Max Area = Lw = (425/2)(425/6) = 425^2/12 = about 180625/12 = about 15,052 square feet

(no guarantees these calculations are error free, but it looks about right)

Answer 2

Draw and label a diagram!

2L + 6W = 850

2L = 850 - 6W

L = 425 - 3W

Area = W(425 - 3W)

Can you transform, derive first drivative, set equal to zero, solve, and answer?