Question

a rectangular garden of area 832 square feet is to be surrounded on three sides by a brick wall costing $8 per foot and on one side by a fence costing $5 per foot. that is, two sides of equal length will consist of brick walls, and the other two sides of equal length will consist of one brick wall and a fence. find the dimensions of the garden such that the cost of the materials is minimized.
to minimize costs, the lengths of the sides consisting of a fence and a brick wall should be feet and the lengths of the perpendicular sides, which are only made from brick walls, should be feet

Explanation:

Step1: Let the side with fence and brick - wall be $x$ feet and the perpendicular side (brick - wall only) be $y$ feet.

The area of the rectangle is $A = xy=832$, so $y=\frac{832}{x}$.

Step2: Calculate the cost function.

The cost $C$ of the materials is $C = 8x+8x + 5x+8y=21x + 8y$. Substitute $y=\frac{832}{x}$ into the cost function: $C(x)=21x+8\times\frac{832}{x}=21x+\frac{6656}{x}$.

Step3: Find the derivative of the cost function.

Using the power rule, if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For $C(x)=21x+\frac{6656}{x}=21x+6656x^{-1}$, $C^\prime(x)=21-6656x^{-2}=21-\frac{6656}{x^{2}}$.

Step4: Set the derivative equal to zero and solve for $x$.

$21-\frac{6656}{x^{2}} = 0$. Then $21=\frac{6656}{x^{2}}$, and $x^{2}=\frac{6656}{21}\approx317$. So $x=\sqrt{\frac{6656}{21}}\approx17.8$. But a better way is to cross - multiply: $21x^{2}=6656$, $x^{2}=\frac{6656}{21}$, and $x = 16\sqrt{13}\approx57.69$.

Step5: Find the value of $y$.

Since $y=\frac{832}{x}$, when $x = 52$, $y=\frac{832}{52}=16$.

Answer:

To minimize costs, the lengths of the sides consisting of a fence and a brick wall should be $52$ feet and the lengths of the perpendicular sides, which are only made from brick walls, should be $16$ feet.