To find the dimensions of a box with a square base and an open top that has a volume of 32,000 cm³ while minimizing the surface area (and thus the amount of material used), we can follow these steps:
Step 1: Define the Variables
Let the length of one side of the square base be x (in cm), and let the height of the box be h (in cm).
Step 2: Set Up the Volume Equation
The volume V of the box can be defined as:
V = x2 * h
Since we want the volume to be 32,000 cm³, we set up the equation:
x2 * h = 32000
Step 3: Express Height in Terms of Base Width
From the volume equation, we can express height h in terms of x:
h = rac{32000}{x2}
Step 4: Set Up the Surface Area Equation
The surface area S of the box (which we want to minimize) can be written as:
S = x2 + 4xh
Substituting the expression for h from the volume equation, we have:
S = x2 + 4x * rac{32000}{x2} = x2 + rac{128000}{x}
Step 5: Differentiate and Find Critical Points
To minimize the surface area, we need to find the derivative of S with respect to x and set it to zero:
rac{dS}{dx} = 2x - rac{128000}{x2}
Setting the derivative equal to zero gives:
2x - rac{128000}{x2} = 0
Multiplying through by x2 to eliminate the fraction:
2x3 - 128000 = 0
Solving for x:
x3 = 64000
x =
oot{3}{64000} = 40 ext{ cm}
Step 6: Calculate h
Now that we have the value of x, we can find h:
h = rac{32000}{x2} = rac{32000}{402} = rac{32000}{1600} = 20 ext{ cm}
Step 7: Conclusion
Thus, the dimensions of the box that minimize the amount of material used are:
- Base length (x): 40 cm
- Height (h): 20 cm
These dimensions will give you a volume of 32,000 cm³ while minimizing the surface area of the box, making it efficient in terms of material usage.