Respuesta :
Answer:
The dimensions are: length = 16 feet, width = 16 feet and height = 12 feet
or Dimensions are 16 ft *16 ft *12 ft
Step-by-step explanation:
Given:
Volume of rectangular with square base is 3072 cubic feet.
Let "x" be the side of the square and "h" be the height of the rectangular.
Volume of the rectangular (V) = area of the base * height
Now plug in the given value V = 3072
3072 = [tex]x^2 *h[/tex] -------------------(1)
Let's find the area of the top.
Which is the square. The area of the top = [tex]x^2[/tex]
The area of the bottom is the same = [tex]x^2[/tex]
The area of the side = x*h
There are 4 sides, therefore, the areas of the 4 sides = 4xh
Now let's find the total cost.
Given: The material for the top and sides costs $4 per square foot and the material for the bottom costs $2 per square foot.
Total cost (T)= [tex]4x^2 +4*4xh + 2x^2\\= 4x^2 + 16xh + 2x^2\\= 6x^2 + 16xh[/tex]
Now let's find h from the equation (1)
h = [tex]\frac{3072}{x^2}[/tex]
Now plug in h = [tex]\frac{3072}{x^2}[/tex] in Total cost (T), we get
T = [tex]6x^2 + 16x(\frac{3072}{x^2} )[/tex]
Simplifying the above, we get
T = [tex]6x^2 + \frac{49152}{x}[/tex]
To find the minimize the total cost, we need to find the derivative of T with respect to x.
T'(x) = 12x - [tex]\frac{49152}{x^2}[/tex]
Now set the derivative equal to zero and find the value of x.
12x - [tex]\frac{49152}{x^2}[/tex] = 0
[tex]\frac{12x^3 - 49152}{x^2} = 0[/tex]
[tex]12x^3 - 49152 = x^2.0\\12x^3 - 49152 = 0\\12x^3 = 49152\\[/tex]
Dividing both sides by 12, we get
[tex]x^3 = 4096[/tex]
Taking the cube root on both sides, we get
x = 16
Therefore, the length and the width of the base is 16
Now we have to find height.
height = [tex]\frac{3072}{x^2}[/tex]
Now plug in x = 16 in the above height, we get
height = [tex]\frac{3072}{16^2} \\= \frac{3072}{256} \\= 12[/tex]
Therefore, the height of the rectangular shipping crate is 12 feet.
So, the dimensions of the crate that will minimize the total cost of material are length = 16 feet, width = 16 feet and height = 12 feet.
