A rectangular garden next to a building is to be fenced on three sides. Fencing for the side parallel to the building costs $60 per foot, and material for the other two sides costs $20 per foot. If $1,500 is to be spent on fencing, what are the dimensions of the garden with the largest possible area?
A square will give the largest possible area.
60x + 20x + 20x = 1,500
Solve for x.
100x=1500
x=1500/100
x=15
Let's assume the length of the garden parallel to the building is L and the width is W.
To fence the side parallel to the building, it costs $60 per foot, and the two sides perpendicular to the building cost $20 per foot. So, the total cost of fencing can be calculated as follows:
Total Cost = Cost of fencing side parallel to the building + Cost of fencing the other two sides
Total Cost = 60L + 20(2W)
1500 = 60L + 40W
To find the dimensions of the garden with the largest possible area, we need to maximize the area. The area of a rectangle is given by:
Area = Length × Width
We can express the length in terms of the width by rearranging the equation:
60L = 1500 - 40W
L = (1500 - 40W)/60
Substituting this expression for L in the area equation:
Area = (1500 - 40W)/60 × W
Area = (1500W - 40W^2)/60
Area = (25W - (2/3)W^2)
To find the maximum area, we can take the derivative of the area equation with respect to W, and set it equal to zero:
d(Area)/dW = 25 - (4/3)W
25 - (4/3)W = 0
25 = (4/3)W
W = (3/4) × 25 = 18.75
Substituting W = 18.75 back into the length equation:
L = (1500 - 40 × 18.75)/60
L ≈ 23.44
Therefore, the dimensions of the garden with the largest possible area are approximately 23.44 feet (length) by 18.75 feet (width).
To find the dimensions of the garden with the largest possible area, we will need to use optimization techniques by representing the problem mathematically and then solving it.
Let's assume that the length of the garden parallel to the building is 'x' feet and the width is 'y' feet. We need to maximize the area, so the objective function will be A = x*y.
The given information states that fencing for the side parallel to the building costs $60 per foot, and the other two sides cost $20 per foot. This allows us to determine the cost function, C, as follows:
C = 60x + 20y + 20y
C = 60x + 40y
We know that the total cost of fencing is $1,500, so we can set up the equation:
60x + 40y = 1500
Now, we need to express one of the variables (y or x) in terms of the other to establish an equation with a single variable. Let's solve for y:
40y = 1500 - 60x
y = (1500 - 60x) / 40
y = (375 - (3/2)x)
We can substitute this expression for y into the area equation A = x*y:
A = x * [(375 - (3/2)x)]
A = 375x - (3/2)x^2
To find the maximum value of A, we will differentiate A with respect to x:
dA/dx = 375 - (3/2) * 2x
dA/dx = 375 - 3x
Setting this derivative equal to zero, we solve for x:
375 - 3x = 0
3x = 375
x = 375 / 3
x = 125
Now that we have the value of x, we can substitute it back into the equation for y:
y = (375 - (3/2) * 125)
y = (375 - 187.5)
y = 187.5
Therefore, the dimensions of the garden with the largest possible area can be determined as follows:
Length (x) = 125 feet
Width (y) = 187.5 feet