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 length of the side with fence and brick - wall be $x$ and the length of the perpendicular side (brick - wall only) be $y$.

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 $C=(8 + 5)x+2\times8y=13x + 16y$.
Substitute $y=\frac{832}{x}$ into the cost function: $C(x)=13x+16\times\frac{832}{x}=13x+\frac{13312}{x}$.

Step3: Find the derivative of the cost function.

Using the power rule, if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For $C(x)=13x+\frac{13312}{x}=13x+13312x^{-1}$, $C^\prime(x)=13-13312x^{-2}=13-\frac{13312}{x^{2}}$.

Step4: Set the derivative equal to zero to find critical points.

$13-\frac{13312}{x^{2}} = 0$.
Add $\frac{13312}{x^{2}}$ to both sides: $13=\frac{13312}{x^{2}}$.
Cross - multiply: $13x^{2}=13312$.
Divide both sides by 13: $x^{2}=\frac{13312}{13}=1024$.
Take the square root of both sides: $x = 32$ (we consider the positive value since length cannot be negative).

Step5: Find the value of $y$.

Since $y=\frac{832}{x}$, when $x = 32$, $y=\frac{832}{32}=26$.

Answer:

To minimize costs, the lengths of the sides consisting of a fence and a brick wall should be 32 feet and the lengths of the perpendicular sides, which are only made from brick walls, should be 26 feet.