A rectangular garden of area 480 square feet is to be surrounded on three sides bya brick wall costing $12 per foot and on one side by a fence costing $5 per foot. Find the dimensions of the garden such that the cost of the materials is minimized.
To find the dimensions of the garden such that the cost of the materials is minimized, we need to set up an equation for the total cost and then find the minimum point of that equation.
Let's call the length of the rectangular garden L (in feet) and the width W (in feet).
The area of the rectangular garden is given as 480 square feet, so we have the equation:
L * W = 480
To find the cost, we need to consider the materials on each side of the garden.
Three sides of the garden are surrounded by a brick wall, and the cost of the brick wall is $12 per foot. So, the cost of the brick wall is:
Brick Wall Cost = 3L * 12
One side of the garden is surrounded by a fence, and the cost of the fence is $5 per foot. So, the cost of the fence is:
Fence Cost = W * 5
The total cost of the materials is the sum of the Brick Wall Cost and the Fence Cost:
Total Cost = Brick Wall Cost + Fence Cost = 3L * 12 + W * 5
Now, we can substitute the value of W in terms of L using the equation for the area of the garden:
W = 480 / L
Substituting this into the equation for the Total Cost:
Total Cost = 3L * 12 + (480 / L) * 5
To find the minimum point, we take the derivative of the Total Cost equation with respect to L, set it equal to zero, and solve for L:
d(Total Cost)/dL = 0
Differentiating the Total Cost equation:
d(Total Cost)/dL = 36 - (2400 / L^2)
Setting it to zero:
36 - (2400 / L^2) = 0
Multiplying through by L^2:
36L^2 - 2400 = 0
Dividing through by 12:
3L^2 - 200 = 0
Now, we can solve this quadratic equation for L using the quadratic formula:
L = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, a = 3, b = 0, and c = -200.
L = (-0 ± sqrt(0^2 - 4 * 3 * (-200)) / (2 * 3)
L = (± sqrt(2400)) / 6
L ≈ ± 15.49 / 6
Taking the positive value:
L ≈ 15.49 / 6 ≈ 2.581
Substituting this value of L back into the equation for the width:
W = 480 / L
W ≈ 480 / 2.581 ≈ 186.31
Therefore, the dimensions of the garden that minimize the cost of materials are approximately L = 2.581 feet and W = 186.31 feet.