Given: available material= 1200cm^2

Box w/ square base & open top
Find: Largest possible value of the box

Can anyone give me a detailed, step-by-step explanation for this problem. I'm studying for an exam and have been working on this for awhile.

The help would be very much appreciated!

To find the largest possible value of the box with a square base and open top, we need to optimize the volume of the box given the constraint that the available material is 1200cm^2.

Let's break down the problem into steps:

Step 1: Define the variables
Let's assume the side length of the square base is 'x' cm.

Step 2: Define the equations
First, we need to find the surface area of the box for optimization.
The surface area of the base (a square) is given by: B = x^2 cm^2
The surface area of the four sides (all rectangles) is given by: S = 4xh cm^2. Note that 'h' represents the height of the box.

Given that the available material is 1200cm^2, we can form the equation:
B + S = 1200

Step 3: Simplify the equation
Substituting the formulas for B and S into the equation, we get:
x^2 + 4xh = 1200

Step 4: Isolate 'h' in terms of 'x'
Rearrange the equation to isolate 'h':
4xh = 1200 - x^2
h = (1200 - x^2) / 4x

Step 5: Define the volume equation
The volume of the box is given by:
V = Bh
Substituting the formulas for B and h, we have:
V = x^2 * [(1200 - x^2) / 4x]

Step 6: Simplify the volume equation
Simplify the equation by canceling out common factors and expanding:
V = (1/4)x(300-x^2)
V = 75x - (1/4)x^3

Step 7: Find the derivative of the volume equation
To find the maximum volume, we need to find the critical points of the volume equation. We can do this by finding the derivative of the volume equation and setting it equal to zero:
dV/dx = 75 - (3/4)x^2

Step 8: Set the derivative equal to zero and solve for 'x'
Setting the derivative equal to zero, we have:
75 - (3/4)x^2 = 0
Multiply both sides by 4 to eliminate the fraction:
300 - 3x^2 = 0
Rearrange the equation:
3x^2 = 300
Divide both sides by 3:
x^2 = 100
Take the square root of both sides:
x = ±10

However, since we are considering the side length of the square base, 'x' cannot be negative. Therefore, x = 10 cm.

Step 9: Calculate the corresponding volume
Substitute the value of 'x' back into the volume equation:
V = 75x - (1/4)x^3
V = 75(10) - (1/4)(10)^3
V = 750 - 250
V = 500 cm^3

Therefore, the largest possible value of the box is 500 cm^3, and the side length of the square base is 10 cm.