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.
To find the largest possible volume of the box, we need to maximize the volume while using the given amount of material.
Let's break down the problem and find equations to help solve it step by step:
Step 1: Define the variables:
Let's suppose the side length of the square base is "x" centimeters, and the height of the box is "h" centimeters.
Step 2: Determine the constraints:
The total surface area of the box consists of the area of the base and the four sides. Given that the material available is 1700 square centimeters, we can write the equation:
4x^2 + x^2 = 1700
Simplifying, we get:
5x^2 = 1700
Step 3: Solve for x:
Divide both sides of the equation by 5:
x^2 = 340
Taking the square root of both sides, we find:
x = √340
x ≈ 18.44 (rounded to two decimal places)
Step 4: Find the height:
To find the height "h," we can use the available material and subtract the area of the square base. The equation will be:
2xh = 1700 - x^2
Substituting the value of x:
2(18.44)h = 1700 - 340
36.88h = 1360
h ≈ 36.83 (rounded to two decimal places)
Step 5: Calculate the maximum volume:
The volume of the box is given by V = x^2h. Substituting the values for x and h:
V ≈ (18.44)^2 * 36.83
V ≈ 12123.51
Therefore, the largest possible volume of the box is approximately 12123.51 cubic centimeters.