A rectangular garden of area 300 square feet is to be surrounded on three sides by a brick wall costing $\$ 10$ 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 $\square$ feet and the lengths of the perpendicular sides, which are only made from brick walls, should be feet.
Answer
Final Answer: To minimize costs, the lengths of the sides consisting of a fence and a brick wall should be \(\boxed{20}\) feet and the lengths of the perpendicular sides, which are only made from brick walls, should be \(\boxed{15}\) feet.
Step 1 ：Let the length of the side that will be surrounded by a fence be denoted as \(x\), and the length of the side that will be surrounded by a brick wall be denoted as \(y\). The area of the garden is given by \(x \cdot y = 300\).
Step 2 ：The cost of the materials is given by \(C = 10 \cdot (2y + x) + 5 \cdot x\), which we want to minimize.
Step 3 ：We can express \(y\) in terms of \(x\) using the area constraint, and then substitute this into the cost function to get a function of one variable. So, \(y = 300/x\) and \(C = 15 \cdot x + 6000/x\).
Step 4 ：To find the minimum cost, we need to find the derivative of the cost function and set it equal to zero. So, \(C' = 15 - 6000/x^2\). The critical points are \(x = -20\) and \(x = 20\).
Step 5 ：To determine whether these critical points are minimums or maximums, we can use the second derivative test. The second derivative of the cost function is \(C'' = 12000/x^3\).
Step 6 ：Substituting the critical points into the second derivative, we find that \(C''(-20) < 0\) and \(C''(20) > 0\). Therefore, the cost function reaches a minimum at \(x = 20\).
Step 7 ：This means that the length of the side that will be surrounded by a fence should be 20 feet. We can substitute \(x = 20\) into the equation \(y = 300/x\) to find the length of the side that will be surrounded by a brick wall, which is \(y = 15\) feet.
Step 8 ：Final Answer: To minimize costs, the lengths of the sides consisting of a fence and a brick wall should be \(\boxed{20}\) feet and the lengths of the perpendicular sides, which are only made from brick walls, should be \(\boxed{15}\) feet.
