If 6075 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=??? cubic cm
Use the same procedure explained in previous problems of this type, and show your work.
You have two variables, box base length a and height h. I will assume cutout corner square pieces are not used; other wiseyou will need a lot of tape to make sides from cut corners.
(a + 2h)^2 = 6075
The box will be made by folding up the four protruding ends of a fat looking cross, made by cutting square corners off a square that has side length a + 2h.
Volume = a^2*h
Express volume V in terms of a or h only and set the derivative with respect to the remaining variable equal to zero.
Compute the two variables and the volume
im stuck
To find the largest possible volume of the box, we need to optimize the dimensions of the box.
Let's assume that the side length of the square base is "x" centimeters. Since the box has a square base, its height will also be "x" centimeters.
Now, to find the volume of the box, we use the formula:
Volume = Base Area * Height
The base area of the square is given by:
Base Area = side length * side length = x * x = x^2 square centimeters.
Since the box is open at the top, the material is only used to construct the base and the sides. The total surface area of the box (including the base and the four sides) is given by:
Surface Area = Base Area + 4 * Side Area
The side area of each side is given by multiplying the height by the side length, which gives us:
Side Area = x * x = x^2 square centimeters.
So, the surface area of the box is:
Surface Area = x^2 + 4(x^2) = x^2 + 4x^2 = 5x^2 square centimeters.
We are given that the total material available is 6075 square centimeters. Therefore, we can equate the surface area to the total material available:
5x^2 = 6075
Now, let's solve this equation for "x".
Divide both sides of the equation by 5:
x^2 = 1215
Take the square root of both sides to isolate "x":
x = √(1215)
Now, calculate the square root of 1215:
x ≈ 34.9 centimeters
Now that we have the value of "x", we can substitute it back into the volume equation to find the largest possible volume:
Volume = Base Area * Height = x^2 * x = (34.9)^2 * 34.9 = 42,320.61 cubic centimeters (approximately).
Therefore, the largest possible volume of the box is approximately 42,320.61 cubic centimeters when the side length of the square base is approximately 34.9 centimeters.