Respuesta :
Answer:
Dimensions of the container should be 12×12×7.5 ft to minimize the making cost.
Step-by-step explanation:
A trash company is designing an open top, rectangular container having volume = 1080 ft³
Let the length of container = x ft , width of the container = y ft and height of the container = z ft.
So volume of the rectangular container = xyz = 1080 ft³
Or [tex]z=\frac{1080}{xy}[/tex] ft -----(1)
Cost of making the bottom of the container = $5 per square ft
Area of the bottom = xy
Cost of making the bottom @ $5 per square ft = 5xy
Area of all sides of the container = 2(xz + yz) = 2z(x+ y)
Now it has been given that cost of making all sides of the container is = $4 per square ft
So total cost to manufacture sides = 4[2z(x + y)]
Now cost of making bottom and sides of the container = 5xy + 8z(x + y)
We put the value of z from equation 1
Total cost A = 5xy+8(x + y)[tex](\frac{1080}{xy})[/tex]
A = 5xy +[tex]8(\frac{1080}{y})+8(\frac{1080}{x})[/tex]
Now we will find the derivative of A and equate it to the zero
[tex]\frac{dA}{dx}=0[/tex] and [tex]\frac{dA}{dy}=0[/tex]
[tex]\frac{dA}{dx}=5y+8(1080)(0)+8(1080)(-\frac{1}{y^{2}})=0[/tex]
5y =[tex]\frac{8\times1080}{y^{2} }[/tex]
5y³ = 8640
y³ =[tex]\frac{8640}{5}=1728[/tex]
y = 12 ft
For [tex]\frac{dA}{dy}=0[/tex]
[tex]\frac{dA}{dy}=5x+\frac{8(-1080)}{x^{2}}[/tex]=0
5x =[tex]\frac{8(1080)}{x^{2} }[/tex]
5x³ = 8640
x³ = 1728
x = 12
Now from equation 1
z =[tex]\frac{1080}{x}[/tex]
=[tex]\frac{1080}{144}[/tex]
z = 7.5
Therefore, dimensions of the container should be 12×12×7.5 ft to minimize the making cost.
