A fence is to be built to enclose a rectangular area of 800 square feet. The fence along three sides is to be made of material that costs $6 per foot. The material for the fourth side costs $18 per foot. Find the dimensions of the rectangle that will allow for the most economical fence to be built
Let's assume the dimensions of the rectangle are x and y, where x is the length and y is the width.
The area of the rectangle is given by the equation xy=800.
We need to minimize the cost of the fence, which is given by the equation Cost=3(6x)+18y
Simplifying this equation gives Cost=18x+18y.
We can solve for y in terms of x using the area equation: y=800/x.
Substituting this into the cost equation gives Cost=18x+18(800/x)).
To find the dimensions that will minimize the cost, we need to find the derivative of the cost equation with respect to x and set it equal to zero:
d(Cost)/dx=18-14400/x^2=0.
Solving this equation gives x=40.
Substituting this value back into the area equation gives y=800/40=20.
Therefore, the dimensions of the rectangle that will allow for the most economical fence to be built are 40 feet by 20 feet.