If 507 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 determine the dimensions of the box that will maximize its volume.
Let's break down the problem into smaller steps:
Step 1: Define variables
Let's assume that the side length of the square base is "x" centimeters. We can use this information to determine the dimensions of the box.
Step 2: Calculate the surface area of the box
The box has four sides and a base. Since the box is open at the top, there is no top surface. We can calculate the surface area by summing the areas of the sides and the base. The base area is x * x = x^2 square centimeters, and the side area is 4x * h, where "h" is the height.
Therefore, the total surface area of the box is: x^2 + 4xh.
Step 3: Express the height in terms of x
The total surface area of the box is given as 507 square centimeters. Plugging in x^2 + 4xh = 507, we can express the height "h" in terms of "x."
Step 4: Express the volume of the box in terms of x
The volume of the box is the product of the base area (x^2) and the height "h." Therefore, the volume V can be expressed as V = x^2 * h.
Step 5: Express the volume in terms of x alone
Using the expression for "h" from step 3, substitute this into the volume equation in step 4. Now we have the volume expressed in terms of x alone.
Step 6: Find the critical points
To maximize the volume, we need to find the critical points by taking the derivative of the volume equation with respect to "x" and setting it equal to zero.
Step 7: Determine the maximum volume
From the critical points found in step 6, evaluate the volume function at each point. The largest value obtained will be the maximum volume of the box.
This process involves equations and calculations that need to be solved step by step. Let's perform these calculations and find the largest possible volume of the box.