Question

if 2400 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Answer 1

The largest possible volume of the box can be found by maximizing the volume of a cube with a square base. The surface area of the cube is equal to the available material, which is 2400 square centimeters. Solving for the side length of the square base, we find that it is 20 centimeters.

Step-by-step explanation:

The largest possible volume of the box can be found by maximizing the volume of a cube with a square base. Let x be the length of one side of the square base and h be the height of the box.

The surface area of the cube is given by 6x². Since the box has an open top, the surface area is equal to the available material, which is 2400 square centimeters. So, 6x² = 2400.

Solving for x, we have x² = 400, and taking the square root of both sides, x = 20. Therefore, the largest possible volume of the box is V = x²h = 20²h = 400h cubic centimeters.

Answer 2

The largest possible volume of the box, obtained from the maximum value of the function for the volume of the box is about 11,313.71 cubic centimeters

The steps used to find the largest volume of the box are as follows;

Let x represent the sid]de length of the square base, and let y represent the height of the box

The surface area is; x² + 4·x·y

The volume is; V = x²·y

We get;

x² + 4·x·y = 2,400

y = (2,400 - x²)/4·x

V = x² × ((2,400 - x²)/4·x)

x² × ((2,400 - x²)/(4·x) = 600·x - x³/4

V = 600·x - x³/4

dV/dx = d(600·x - x³/4)/dx

d(600·x - x³/4)/dx = 600 - 3·x²/4

When the volume is maximum or minimum, we get;

dV/dx = 0, therefore; 600 -3·x²/4 = 0

600 = 3·x²/4

3·x²/4 = 600

x² = (600/3) × 4

(600/3) × 4 = 800

x² = 800

x = √(800)

x = 20·√2

V = 600·x - x³/4

V = 600 × (20·√2) - (20·√2)³/4

600 × (20·√2) - (20·√2)³/4 ≈ 11,313.71

The largest possible volume of the box is V ≈ 11,313.71 cm³