Question

A rectangular parking area measuring 6000 ft squared is to be enclosed on three sides using​ chain-link fencing that costs ​$5.50 per foot. The fourth side will be a wooden fence that costs ​$6 per foot. What dimensions will minimize the total cost to enclose this​ area, and what is the minimum​ cost?

Answer 1

Answer:

length x =79.2 ft

width y = 75.75 ft

minimum​ cost P = Rs 1742.41212

Step-by-step explanation:

Let the length of the parking area be 'x', and the width  be 'y'.

Then, we can write the following equations:

Then,  Area of the park: A =x×y = 6000

Now,  Price of the fences P = 2×5.5x + 5.5y + 6y

P = 11x + 11.5y

From the first equation, we have that y = 6000/x

Using this value in the equation for P, we have:

P = 11x + 11.5×6000/x = 11x + 69000/x

To find the minimum of this function, we need to take its derivative and then make it equal to zero:

⇒ dP/dx = 11 - 69000/x^2 = 0

⇒x^2 = 69000/11

x  = 79.20 ft

This is the x value that gives the minimum cost.

Now, finding y and P, we have:

x×y = 6000

y = 6000/79.2 = 75.75 ft

P = 11x + 69000/x = Rs 1742.41212