Question

A fence is to be built to enclose a rectangular area. The fence along three sides is to be made of material that costs $5 per foot. The material for the fourth side costs $15 per foot.If $3,000 is available for the fencing, find the dimensions of the rectangle that will enclose the most area.

Answer 1

Step 1)

As given the fence along three sides is to be made of material that costs $5 per foot. The material for the fourth side costs $15 per foot

Hence we can write cost function is given by,

As given $3,000 is available for the fencing hence we can write,

-----------------------------------------------------1)

Step 2)

we know that area of the rectangular field with sides x and y is given by,

we have,

Hence,

Hence,

equate it to 0 we can say that,

Hence x = 150 is the critical point of area function

Step 2)

we have,

Hence,

Hence,

As A''(150) = -1 < 0 according to second derivative test we can say that x = 150 is a maximum point

Hence we can say that area of the rectangular field is maximum at x = 150 ft

we have,

Hence,

Hence we can say that area of the rectangular field is maximum if x = 150 ft and y = 75 ft

we can write dimensions of the rectangle that will enclose the most area is x = 150 ft and y = 75 ft