If 1700 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.
Volume = ?
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To find the largest possible volume of the box, we need to maximize the volume while considering the constraint of the available material.
Let's assume the side length of the square base of the box is 'x' centimeters. Since the box has a square base and an open top, it would have four sides and a bottom.
The area of the base would be x * x = x^2 square centimeters.
The total area of the four sides would be 4x * h, where 'h' is the height of the box.
The total area of material used is the sum of the base area and the side area, which gives:
Area = x^2 + 4xh
We know that the total area of material available is 1700 square centimeters.
Area = 1700
Substituting the equation for the area, we get:
x^2 + 4xh = 1700
Now, we need to express one variable in terms of the other to find the maximum volume. Let's solve for h:
4xh = 1700 - x^2
h = (1700 - x^2) / 4x
The volume of the box is given by the equation:
Volume = base area * height = x^2 * h = x^2 * [(1700 - x^2) / 4x]
Simplifying further:
Volume = (x^2 * (1700 - x^2)) / 4x
= (1700x - x^3) / 4
To find the largest possible volume, we need to maximize this equation. We can do this by finding the derivative of the equation with respect to 'x' and setting it equal to zero.
d(Volume)/dx = 1700/4 - 3x^2/4
Setting the derivative equal to zero:
1700/4 - 3x^2/4 = 0
1700 - 3x^2 = 0
3x^2 = 1700
x^2 = 1700/3
x = √(1700/3)
Now that we have the value of 'x', we can calculate the maximum volume by substituting it back into the volume equation:
Volume = (x^2 * (1700 - x^2)) / 4
= (√(1700/3))^2 * (1700 - (√(1700/3))^2) / 4
= (1700/3) * (1700 - 1700/3) / 4
= (1700/3) * (2/3) / 4
= (1700 * 2) / (3 * 3 * 4)
= 566.67 cubic centimeters
Therefore, the largest possible volume of the box is approximately 566.67 cubic centimeters.