Given that the area of the playground is 910 square feet, let the length of the playground be x, then the width of the playground is given by [tex] \frac{910}{x} [/tex]
Given that 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 $15 per foot. Let the side with the brick fence be the side which measures x feet.
Then the cost for fencing the entire playground is given by
[tex]C=5\left( \frac{910}{x} \right)+5x+5\left( \frac{910}{x} \right)+15x= \frac{9100}{x} +20x[/tex]
For, minimum cost,
[tex] \frac{dC}{dx} =0 \\ \\ \Rightarrow- \frac{9100}{x^2} +20=0 \\ \\ \Rightarrow20x^2-9100=0 \\ \\ \Rightarrow x^2-455=0 \\ \\ \Rightarrow x=\pm\sqrt{455}[/tex]
But x can't be negative.
Therefore, the length of the brick fence that will minimize the cost of enclosing the playground is 21.33 feet.
