A builder intends to construct a storage shed having a volume of 1000 cubic feet, a flat roof and a rectangular base whose width is three fourths the length. The cost per square foot of the material is $3 for the floor, $5 for the sides and $4 for the roof. What dimensions will minimize the cost?
If the base width is w, the length is 4/3 w
So, the volume, with height h is
v = w(4/3 w)h
h = 1000/(4/3 w^2) = 750/w^2
Now, the cost is
c(w) = 3(4/3 w^2) + 5(2(wh + 4/3 wh)) + 4(4/3 w^2)
= 28/3 w^2 + 17500/w
Now, minimum c will occur when dc/dw = 0
I get w=5∛(15/2)
width = w
length = 4 w/3
height = h
floor area = 4 w^2/3 so floor cost = $4 w^2
side area = h (2 w + 8w/3) = (h/3)(14 w) so side cost =$ (70/3)hw
roof cost = 4(4w^2/3) = $16 w^2 /3
total cost = (1/3) (28 w^2 + 70hw)
but
volume = 1000 = h (4 w^2/3)
so h = 750/w^2
so total cost
c = (1/3) (28 w^2 + 70hw) = (1/3) (28 w^2 + 70*750/w)
3 c = 28 w^2 + 52500/w
minimizw 3 c
d3c/dw = 0 = 56 w -52500/w^2
56 w^3 = 52500
w^3 = 937.5
w = 9.79 etc, check my arithmetic !
I thought it was 3/4L or is that the same as 4 w/3?
To find the dimensions that will minimize the cost, we need to express the cost function in terms of the dimensions of the storage shed.
Let's assume the length of the storage shed is x. According to the problem, the width is three fourths the length, so the width is (3/4)x.
The volume of a rectangular prism is given by the formula V = lwh, where V is the volume, l is the length, w is the width, and h is the height.
In this case, the volume is given as 1000 cubic feet, so we have:
1000 = x * (3/4)x * h
Simplifying this equation, we get:
1000 = (3/4)x^2 * h
Solving for h, we have:
h = 1000 / ((3/4)x^2)
Now, let's express the cost function in terms of the dimensions. The cost consists of the floor, sides, and roof.
The cost of the floor is given by the area of the floor (length times width) multiplied by the cost per square foot. Since the cost per square foot for the floor is $3 and the area is x * (3/4)x = (3/4)x^2, the cost of the floor is:
3 * (3/4)x^2 = (9/4)x^2
The cost of the sides is given by the area of the sides (width times height) multiplied by the cost per square foot. Since the cost per square foot for the sides is $5 and the area is (3/4)x * h, the cost of the sides is:
5 * (3/4)x * h = (15/4)x * h
The cost of the roof is given by the area of the roof (length times width) multiplied by the cost per square foot. Since the cost per square foot for the roof is $4 and the area is x * (3/4)x = (3/4)x^2, the cost of the roof is:
4 * (3/4)x^2 = (12/4)x^2
Now, we can express the total cost function C in terms of the dimensions:
C = (9/4)x^2 + (15/4)x * h + (12/4)x^2
Substituting the expression for h we found earlier:
C = (9/4)x^2 + (15/4)x * (1000 / ((3/4)x^2)) + (12/4)x^2
Simplifying this equation, we get:
C = (9/4)x^2 + (3750/x) + (12/4)x^2
C = (21/4)x^2 + (3750/x)
Now, to minimize the cost, we can take the derivative of C with respect to x and set it to 0:
dC/dx = (42/4)x - 3750/x^2 = 0
Multiplying through by 4x^2 to get rid of the denominators, we get:
42x^3 - 3750 = 0
Dividing through by 42, we get:
x^3 - 89.3 = 0
Solving for x, we find x ≈ 4.49.
Therefore, the dimensions that will minimize the cost are approximately:
Length: 4.49 feet
Width: (3/4) * 4.49 = 3.37 feet
Height: 1000 / ( (3/4)*(4.49)^2 ) = 59.36 feet
Please note that the dimensions have been rounded to two decimal places.