A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $30/ft and on the other three sides by a metal fence costing $10/ft. If the area of the garden is 42 square feet, find the dimensions of the garden that minimize the cost.
To find the dimensions of the garden that minimize the cost, we need to set up and solve an optimization problem.
Let's suppose the length of the garden is L and the width is W.
According to the problem, the area of the garden is given as 42 square feet, so we have the equation:
L * W = 42
We need to minimize the cost, so our goal is to minimize the total cost C.
The cost of the brick wall is $30 per foot if it encloses one side (length L), which gives us a cost of 30 * L.
The cost of the metal fence is $10 per foot if it encloses the other three sides (length 2W + L), which gives us a cost of 10 * (2W + L).
Therefore, our total cost equation is:
C = 30L + 10(2W + L) = 30L + 20W + 10L = 40L + 20W
Now, we can substitute the area equation into the total cost equation to get a cost equation in terms of only one variable, either L or W.
L * W = 42 --> L = 42 / W
Substituting this back into the cost equation, we have:
C = 40L + 20W = 40(42 / W) + 20W = 1680 / W + 20W
To minimize the cost, we need to find the minimum point of the cost function. We can do this by taking the derivative of the cost function with respect to W, setting it equal to zero, and solving for W.
dC/dW = -1680 / W^2 + 20 = 0
-1680 / W^2 = -20
W^2 = 1680 / 20
W^2 = 84
W = √84 = 9.165 feet (approximately)
Now, we can substitute this value of W back into the area equation to find L:
L = 42 / W
L = 42 / 9.165
L ≈ 4.586 feet (approximately)
So, the dimensions of the garden that minimize the cost are approximately 4.586 feet in length and 9.165 feet in width.
To find the dimensions of the garden that minimize the cost, we need to consider the cost function.
Let's assume the length of the garden is L and the width is W. Since the garden is rectangular, the area is given by L * W.
The cost of the brick wall is $30 per foot, and since it is only on one side, the length of the wall is L.
The cost of the metal fence is $10 per foot, and since it is on the other three sides, the total length is (2 * L) + (2 * W).
To calculate the cost, we can multiply the lengths by their respective costs and sum them up.
Cost = (L * $30) + ((2 * L) + (2 * W)) * $10
Now, we need to express one variable in terms of the other to solve this problem.
The area of the garden is given as 42 square feet, so we have L * W = 42.
From this equation, we can express one variable in terms of the other. Let's solve for L:
L = 42 / W
Substituting this value of L into the cost equation, we get:
Cost = (42 / W * $30) + ((2 * (42 / W)) + (2 * W)) * $10
Now we have the cost equation in terms of W only. To find the minimum cost, we can take the derivative of the cost with respect to W and set it equal to zero, and solve for W.
d(Cost) / dW = 0
By simplifying and solving this equation, we can find the value of W that minimizes the cost.
Once we have the value of W, we can substitute it back into the equation L = 42 / W to find the corresponding value of L.
Therefore, to find the dimensions of the garden that minimize the cost, we need to solve the derivative equation and substitute the obtained values of W and L into the area equation.
Let x be the brick wall side length and y be the other dimension.
x y = 42 (area requirement)
10 (x+2y) + 30 x = Cost(x,y)
Substitute 42/x for y in the Cost equation to get an equation for cost in terms of x only.
Cost(x) = 10(x + 84/x) + 30x
= 40x + 84/x
Set the derivative equal to 0 and solve fpr x to see where the minimum cost occurs.