ABC Daycare wants to build a fence to enclose a rectangular playground. The area of the playground is 950 square feet. The fence along three of the sides costs $5 per foot and the fence along the fourth side, which will be made of brick, costs $10 per foot. Find the length of the brick fence that will minimize the cost of enclosing the playground. (Round your answer to one decimal place.)
Let x be the length of the brick side (and one fenced side)
Let y be the width of the other two sides.
Then we have
xy = 950
The cost c is then
c = 10x + 5(x+2y) = 15x + 10y = 15x + 9500/x
dc/dx = 15 - 9500/x^2
So, find x where dc/dx = 0
To solve this problem, we need to minimize the cost of enclosing the playground by finding the length of the brick fence.
Let's assume the length of the rectangular playground is represented by "l" and the width is represented by "w".
Given that the area of the playground is 950 square feet, we have the equation: l * w = 950.
To minimize the cost, we need to determine the length of the brick fence (which is the fourth side of the rectangle). Let's label this side as "x".
The total cost is a combination of the cost of the three sides made of regular fence (which costs $5 per foot) and the cost of the brick fence (which costs $10 per foot).
The cost of the three regular fences is: 2lw + w * 5.
The cost of the brick fence is: x * 10.
We can write the total cost equation as follows: Total Cost = 2lw + w * 5 + x * 10.
Since we are trying to minimize the cost, we want to find the value of x that will give us the minimum total cost.
Now, we can substitute l * w = 950 into the cost equation.
Total Cost = 2(950/w) + 5w + 10x.
To find the minimum cost, we need to find the derivative of the Total Cost equation with respect to x and set it equal to zero.
d(Total Cost)/dx = 0.
Taking the derivative with respect to x, we get:
d(Total Cost)/dx = -10x^2 + a, where a represents a constant that doesn't involve x.
Setting -10x^2 + a = 0, we can solve for x.
-10x^2 = -a.
x^2 = a/10.
x = sqrt(a/10), where sqrt(a/10) represents the square root of a/10.
Since we want to find the length of the brick fence that minimizes the cost, we need to substitute the value of x back into the original equation: l * w = 950.
l * w = 950.
l * w = 950.
lw = 950.
Now, we need to solve for l and w in terms of x.
l = 950/w.
w = 950/l.
Rewriting these equations in terms of x:
l = 950/(sqrt(a/10)).
w = 950/(sqrt(a/10)).
To find the length of the brick fence that minimizes the cost, we need to substitute l and w into the Total Cost equation and solve for x.
Total Cost = 2lw + w * 5 + x * 10.
Substituting the values of l and w:
Total Cost = 2(950/(sqrt(a/10))) * (950/(sqrt(a/10))) + (950/(sqrt(a/10))) * 5 + x * 10.
Simplifying this equation will give us the equation for the Total Cost in terms of x.
Once we have the Total Cost equation, we can differentiate it with respect to x and set it equal to zero to find the value of x that minimizes the cost.
After finding the value of x, we can substitute it back into the equation lw = 950 to find the corresponding values of l and w.
Please note that this is a rough explanation of the process. In real-world scenarios, it's best to use a computational tool or mathematical software to solve the equations and find the minimum cost directly.