A landscape architect wishes 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 $20/ft. If the area of the garden is 2,000 ft2, find the dimensions of the garden that minimize the cost.
?= (ft of brick wall)
?= (ft of length of adgesent side of brick wall)
To find the dimensions of the garden that minimize the cost, we need to consider the cost of the brick wall and the cost of the metal fence separately.
Let's assume that the length of the garden is x and the width is y.
So, the area of the garden is given by xy, which is equal to 2000 ft².
Now, let's consider the cost of the brick wall. The garden is enclosed on one side by a brick wall, which means the length of the brick wall is y ft.
Since the cost of the brick wall is $30 per foot, the total cost of the brick wall is 30y dollars.
The other three sides of the garden are enclosed by the metal fence, and the length of each side is x ft.
Since the cost of the metal fence is $20 per foot, the total cost of the metal fence for all three sides is 3 * 20x = 60x dollars.
The total cost, C, is the sum of the cost of the brick wall and the cost of the metal fence:
C = 30y + 60x.
We need to find the dimensions x and y that minimize the cost C. To do this, we can use the given relationship that the area of the garden is 2000 ft².
From the area equation, we can express y in terms of x:
xy = 2000.
y = 2000/x.
Substituting this expression for y into the cost equation:
C = 30(2000/x) + 60x.
Now we can find the minimum cost by differentiating the cost equation with respect to x, setting it equal to zero, and solving for x:
dC/dx = -60000/x² + 60 = 0.
Simplifying the equation:
-60000/x² + 60 = 0.
Multiplying through by x²:
-60000 + 60x² = 0.
Rearranging the equation:
60x² = 60000.
Dividing through by 60:
x² = 1000.
Taking the square root of both sides:
x = √1000.
Simplifying:
x = 31.62 ft.
Now, we can find y by substituting the value of x back into the equation y = 2000/x:
y = 2000/31.62 = 63.25 ft.
Therefore, the dimensions of the garden that minimize the cost are approximately 31.62 ft by 63.25 ft.