If 600 cm2 of material is available to make a box with a square base and a closed top, find the maximum volume of the box in cubic centimeters. Answer to the nearest cubic centimeter without commas. For example, if the answer is 2,000 write 2000.
If we have base x^2 and height y,
2x^2 + 4xy = 600
v = x^2y = x^2(600-2x^2)/4x
= 150x - x^3/2
Now just find x where v' = 0
0.0
20000
To find the maximum volume of a box with a square base and a closed top, we need to determine the dimensions that will give us the largest possible volume.
Let's assume the length of each side of the square base is x cm. Since the base is square, the area of the base can be calculated as x * x = x^2.
The material available to make the box is 600 cm^2. We need to account for the five sides of the box: the base and the four walls (there is no top).
The area of the four walls can be calculated as 4x * h, where h is the height of the box.
Since there is no top, the total surface area of the box is given by x^2 + 4xh.
We know that the material available is 600 cm^2, so we can write the equation: x^2 + 4xh = 600.
Now, we need to express the height of the box in terms of x, so we can write h in terms of x: h = (600 - x^2) / (4x).
The volume of the box is given by the product of the base area (x^2) and the height (h): V = x^2 * h.
Substituting the expression for h, we get V = x^2 * (600 - x^2) / (4x).
Simplifying the equation, V = (600x - x^3) / 4.
To find the maximum volume, we need to find the value of x that maximizes this equation. We can do this by finding the derivative of V with respect to x, setting it equal to zero, and solving for x.
Taking the derivative, we get dV/dx = (600 - 3x^2) / 4.
Setting dV/dx equal to zero, we have (600 - 3x^2) / 4 = 0.
Simplifying, 600 - 3x^2 = 0.
Rearranging the equation, 3x^2 = 600.
Dividing by 3, we get x^2 = 200.
Taking the square root, we find x = √200 ≈ 14.14.
Since x represents the length of the side of the square base, we will take x = 14 cm, as we are looking for a whole number length.
Now, we can calculate the maximum volume by substituting x = 14 into the equation for V: V = (600 * 14 - (14^3)) / 4.
Evaluating this expression, we find V ≈ 980.
Therefore, the maximum volume of the box is approximately 980 cubic centimeters.