A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $60/ft and on the other three sides by a metal fence costing $50/ft. If the area of the garden is 102 square feet, find the dimensions of the garden that minimize the cost.
Why did the garden hire a landscape architect? Because it needed someone to weed out all the bad ideas!
Let's solve this puzzle, shall we? We have a rectangular garden with one side enclosed by a brick wall, and the other three sides enclosed by a metal fence. The area of the garden is 102 square feet.
Let's say the length of the garden (the side enclosed by the brick wall) is L and the width (the other two sides enclosed by the metal fence) is W.
The perimeter of the garden is P = 2L + W. Since we have three sides enclosed by the metal fence, the total cost of the metal fence is 3W * $50/ft = $150W.
The remaining side enclosed by the brick wall costs $60/ft and has a length L, so the cost of the brick wall is $60L.
Now, let's find an equation that measures the area of the garden in terms of L and W. Since the area of a rectangle is given by A = L * W, we have L * W = 102.
To minimize the cost, we need to minimize the function C(L, W) = $150W + $60L. We can use the equation for the area to express L in terms of W: L = 102/W.
Substituting this into the cost function, we get C(W) = $150W + $60(102/W).
Now we have a function of one variable, W, which represents the width. To find the minimum cost, we need to find the minimum value of this function. We can do this by finding the critical points, where the derivative of the function equals zero.
Differentiating the cost function, we get C'(W) = $150 - $6120/W^2.
Setting C'(W) = 0, we have $150 - $6120/W^2 = 0.
Solving for W, we get W^2 = $6120/$150 = 40.8, which leads to W = √40.8.
Plugging this value back into the equation for L, we have L = 102/√40.8.
Therefore, the dimensions of the garden that minimize the cost are approximately L ≈ 10.1 ft and W ≈ 6.38 ft.
So, the garden's width approximates 6.38 ft, and its length is around 10.1 ft.
To find the dimensions of the garden that minimize the cost, we need to apply calculus techniques.
Let's assume the length of the garden is "L" and the width is "W".
Given that the area of the garden is 102 square feet, we have:
L * W = 102
We want to minimize the cost, which includes the cost of the brick wall and the cost of the metal fence.
The cost of the brick wall is given as $60 per foot, and since the brick wall is enclosing only one side of the garden, the cost of the brick wall is 60 * L.
The cost of the metal fence is given as $50 per foot, and since the metal fence is enclosing the other three sides of the garden, the cost of the metal fence is 50 * (2L + 2W).
Therefore, the total cost is the sum of the cost of the brick wall and the cost of the metal fence:
C = 60L + 50(2L + 2W)
Now, let's express the width "W" in terms of the length "L" by rearranging the area equation:
W = 102 / L
Substitute this expression for "W" into the total cost equation:
C = 60L + 50(2L + 2(102/L))
Simplify the equation:
C = 60L + 100L + 20400/L
C = 160L + 20400/L
To find the minimum cost, we take the derivative of the cost equation with respect to the length "L" and set it equal to 0:
dC/dL = 160 - 20400/L^2 = 0
Now, solve for the length "L":
160 = 20400/L^2
L^2 = 20400/160
L^2 = 127.5
L ≈ √127.5
L ≈ 11.30 ft
Now, substitute the value of "L" back into the width equation to find the width "W":
W = 102 / L
W ≈ 102 / 11.30
W ≈ 9.03 ft
Therefore, the dimensions of the garden that minimize the cost are approximately 11.30 ft by 9.03 ft.
To find the dimensions of the garden that minimize the cost, let's assign variables to the dimensions of the garden.
Let's say the width of the garden is 'w' feet, and the length of the garden is 'l' feet.
We know that the area of the garden is given as 102 square feet, so we can write an equation for the area:
w * l = 102
Now, let's determine the cost equation for enclosing the garden.
The brick wall only encloses one side of the garden, so the cost of the wall is $60 per foot.
The metal fence encloses the other three sides, so the cost of the fence is $50 per foot.
The cost equation can be written as:
Cost = (cost of the wall) + (cost of the fence)
= (60 * l) + (50 * (w + w + l))
Simplifying the cost equation:
Cost = 60l + 100w
Now, we need to express one variable in terms of the other variable.
From the area equation, we can rearrange it to solve for l:
l = 102 / w
Substituting this expression for l in the cost equation:
Cost = 60 * (102 / w) + 100w
Simplifying the cost equation further:
Cost = (6120 / w) + 100w
Now, we have a cost equation in terms of a single variable 'w'. To find the minimum cost, we need to differentiate the cost equation with respect to 'w' and set it equal to zero.
Let's differentiate the cost equation:
d(Cost)/dw = -6120/w^2 + 100
Setting the derivative equal to zero and solving for 'w':
-6120/w^2 + 100 = 0
-6120 + 100w^2 = 0
100w^2 = 6120
w^2 = 61.2
w = √61.2 ≈ 7.82 (rounded to two decimal places)
Now, substitute this value of 'w' back into the area equation to find the value of 'l':
l = 102 / w
l = 102 / 7.82 ≈ 13.05 (rounded to two decimal places)
Therefore, the dimensions of the garden that minimize the cost are approximately 7.82 feet in width and 13.05 feet in length.
let the width be x
and the length y , were the brick wall is along the length.
the xy = 102
y = 102/x
Cost = 50(2x) + 50y + 60y
= 100x + 110y
= 100x + 110(102/x) = 100x + 11220/x
d(cost)/dx = 100 - 11220/x^2
= 0 for a min of cost
11220/x^2 = 100
100x^2 = 11220
x^2 = 11220/100
x = 105.925/10
x = 10.59 ft
they y = 102/x = 9.63 ft
The math tells me that the wall should be build along the shorter side (makes sense)