A rectangular rose garden will be surrounded by a brick wall on three sides and by a fence on the fourth side. The area of the garden will be 1000m^2. The cost of the brick wall is $192/m. The cost of the fencing is $48/m. Find the dimensions of the garden so that the cost of the materials will as low as possible
To find the dimensions of the garden that will minimize the cost of materials, we need to consider the areas and costs of the wall and the fence.
Let's assume that the length of the garden is L meters and the width is W meters.
The area of the garden is given as 1000 m^2, so we can write the equation:
L * W = 1000
We need to minimize the cost of materials, which consists of the cost of the brick wall and the fence. The cost of the brick wall is $192 per meter, and the cost of the fence is $48 per meter.
The cost of the brick wall is determined by the perimeter of the garden, which includes three sides (2 * L + W) multiplied by the cost per meter:
Cost of the brick wall = 192 * (2 * L + W)
The cost of the fence is determined by the remaining side, which is W, multiplied by the cost per meter:
Cost of the fence = 48 * W
To find the dimensions that minimize the cost, we need to express the cost in terms of a single variable. We can express L in terms of W using the area equation:
L = 1000 / W
Substituting this into the cost of the brick wall equation:
Cost of the brick wall = 192 * (2 * (1000 / W) + W)
Now, we can express the total cost, T, in terms of W:
T = Cost of the brick wall + Cost of the fence
= 192 * (2 * (1000 / W) + W) + 48 * W
To find the value of W that minimizes the cost, we can differentiate the total cost equation with respect to W and set it to zero:
dT/dW = 0
Differentiating T with respect to W:
dT/dW = 192 * (-2 * (1000 / W^2) + 1) + 48
= -384000 / W^2 + 192 + 48
= -384000 / W^2 + 240
Setting dT/dW to zero:
-384000 / W^2 + 240 = 0
Simplifying:
384000 / W^2 = 240
Multiplying both sides by W^2:
384000 = 240 * W^2
Dividing both sides by 240:
1600 = W^2
Taking the square root of both sides:
W = 40
Substituting the value of W back into the area equation to find L:
L = 1000 / W
= 1000 / 40
= 25
Therefore, the dimensions of the garden that will minimize the cost of materials are 25 meters by 40 meters.
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 fencing. Let's call the length of the garden L and the width of the garden W.
The garden is surrounded by a brick wall on three sides and a fence on the fourth side. Since we are minimizing the cost, we want to minimize the amount of materials used.
The area of the garden is given as 1000m^2, so we have the equation L * W = 1000.
The cost of the brick wall is $192/m and the cost of the fencing is $48/m. The cost of the brick wall is determined by the perimeter of the garden since it surrounds three sides. The cost of the fencing is determined by the remaining side.
The perimeter of the garden can be calculated as P = 2L + W.
The total cost of the materials is calculated as: Cost = (perimeter of the brick wall * cost of brick wall) + (length of the fence * cost of fence)
Cost = (2L + W) * 192 + W * 48
To minimize the cost, we need to find the values of L and W that minimize the total cost function.
We can use calculus to find the minimum. Taking the derivative of the total cost function with respect to L and W, and setting them equal to 0, we can solve for the values of L and W that minimize the cost.
After solving the equations, we find that L = 20m and W = 50m, or L = 50m and W = 20m.
So, the dimensions of the garden that minimize the cost are either a length of 20m and a width of 50m, or a length of 50m and a width of 20m.
If the brick wall side has length x and the width is y, then
xy = 1000
The cost is
c = 192x + 48(x+2y) = 192x + 48(x + 2000/x) = 240x + 96000/x
to minimize c, we have
dc/dx = 240 - 96000/x^2
dc/dx = 0 at x=20, so the garden is 20 by 50