i am building a rectangular swimming pool. the pool is to have a surface area of 1000 ft.^2. it also has a uniform walk way of 3 ft. surrinding it. X is the lengh of one side of the pool, and Y is te lengh of the other side.

A.) express the area of the plot of land needed for the pool and the surrunding side was as a function of, X.

B.) Determine the dimensions of the plot of land that has the least area.

C.) what is the least area?

Well xy is 1000

Area= pool
Area= (x+3)(y+3) but y=1000/x
1)Area= (x+3)(1000/x +3)

Least area? Take dA/dx, set to zero, and solve for x. Then solve for y.

least area is the area in 1) above with the x you found.

To express the area of the plot of land needed for the pool and the surrounding sidewalk as a function of X, we can use the given information. The area of the pool is 1000 ft², and it has a uniform walkway of 3 ft surrounding it.

Let's denote the length of one side of the pool as X and the length of the other side as Y. Since the pool is rectangular, we have XY = 1000 ft².

To calculate the total area of the plot of land needed, we need to consider the area of the pool and the surrounding sidewalk. The width of the walkway is 3 ft, so the length of the pool with the walkway on one side would be X + 3, and the width of the pool with the walkway on one side would be Y + 3.

Therefore, the area of the plot of land needed for the pool and the surrounding sidewalk can be expressed as a function of X:

A(X) = (X + 3)(Y + 3)

Since XY = 1000, we can rewrite the area function as:

A(X) = (X + 3)(1000/X + 3)

To determine the dimensions of the plot of land that has the least area, we need to find the value of X that minimizes the area function A(X).

To do this, we can take the derivative of A(X) with respect to X and set it equal to zero:

dA/dX = 0

Then, we can solve this equation to find the value(s) of X that minimize the area function. After finding the value(s) of X, we can substitute it back into the area function to determine the dimensions of the plot of land with the least area.

Finally, to find the least area, we substitute the value of X that we found into the area function A(X).

I hope this helps in solving the problem!