This summer Borice decides to put a rectangular swimming pool in his back yard. The material he uses for the border of the length of the pool costs $10 per foot and the material he uses for the width costs $5 per foot. He would like spend $1,000 on the perimeter of the pool. Given these constraints, what are the dimensions of the pool (length and width) that will maximize the surface area of his pool?
length of pool --- x
width of pool -----y
cost = 10x + 5y = 1000
2x + y = 200
y = 200-2x
area = xy
= x(200-2x)
= 200x - 2x^2
d(area)/dx = 200 - 4x
= 0 for a max of area
4x=200
x=50
then y = 200-100 = 100
state your conclusion
To find the dimensions of the pool that will maximize the surface area, we can set up an equation based on the given information.
Let's assume the length of the pool is L feet, and the width is W feet.
First, let's calculate the perimeter of the pool using the given prices. The cost of the length borders would be $10 per foot, and since there are two sides with length L, the cost of the length borders would be 2L * $10 = $20L. Similarly, the cost of the width borders would be 2W * $5 = $10W.
According to the information, the total cost of the borders is $1,000. So the equation can be written as:
$20L + $10W = $1,000
We can simplify this equation by dividing both sides by 10:
2L + W = 100
To maximize the surface area of the pool, we can use the formula: Surface area = Length * Width.
So, the surface area equation becomes:
A = L * W
Now we have two equations that we can use to find the dimensions that maximize the surface area:
2L + W = 100 (Equation 1)
A = L * W (Equation 2)
To solve this problem, we can use a method called substitution. Rearrange Equation 1 to solve for W:
W = 100 - 2L
Now substitute this value of W into Equation 2:
A = L * (100 - 2L)
Simplifying further:
A = 100L - 2L^2
To maximize the surface area, we need to find the value of L that will give us the maximum value of A. This can be done by finding the vertex of the parabola formed by the equation of A.
The equation of a parabola in vertex form is y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.
In this case, let's rewrite the equation of A in the vertex form:
A = -2L^2 + 100L
To find the vertex, we need to determine the values of h and k. The vertex of a parabola occurs at x = -b/2a, where in this case, a = -2 and b = 100. So the x-coordinate of the vertex is:
L = -100 / (2 * -2)
L = 25
Substituting L = 25 into Equation 1:
2(25) + W = 100
W = 50
Therefore, the dimensions of the pool that will maximize the surface area are a length of 25 feet and a width of 50 feet.