What is the maximum volume of a closed box with a square base which can be made by bending the material? The box is to have a surface area of 100in^2.
If the final box has bases x^2 and height h, then
v = x^2 h
and we also have
x^2 + 4xh = 100, so
h = (100-x^2)/4x
That means
v = x^2 (100-x^2)/4x
= x/4 (100-x^2)
= 25x - 1/4 x^3
dv/dx = 25 - 3/4 x^2
maximum volume is achieved when dv/dx=0.
To find the maximum volume of a closed box with a square base made by bending a material, we need to consider the surface area constraint. Here's how you can solve it step by step:
Step 1: Let's assume the length of one side of the square base is 'x' inches.
Step 2: The surface area of a closed box is given by the sum of the areas of its five faces. In this case, since the top and bottom faces are squares with side length 'x', their combined area is 2x^2 square inches.
Step 3: The remaining four faces are rectangles. Two sides of each rectangle will have a length of 'x', and the other two sides will have a length of 'h' (the height of the box). Thus, the combined area of these four faces is 4xh square inches.
Step 4: According to the problem, the total surface area should be 100 in^2. So we have:
2x^2 + 4xh = 100
Step 5: Now we can express 'h' in terms of 'x' by rearranging the equation:
4xh = 100 - 2x^2
h = (100 - 2x^2) / 4x
h = (25 - 0.5x^2) / x
Step 6: The volume of the box is given by the product of the base area (x^2) and the height (h). Thus, the volume (V) is:
V = x^2 * [(25 - 0.5x^2) / x]
V = x * (25 - 0.5x^2)
V = 25x - 0.5x^3
Step 7: To find the maximum volume, we need to find the critical points of the volume function by taking its derivative and setting it equal to zero:
dV/dx = 25 - 1.5x^2 = 0
Step 8: Solve the above equation to find the critical points:
1.5x^2 = 25
x^2 = 25 / 1.5
x^2 = 16.67
x ≈ √16.67
x ≈ 4.08
Step 9: Since we are dealing with dimensions, we take the positive value of 'x'.
Step 10: Substitute the value of 'x' back into the equation for 'h':
h = (25 - 0.5x^2) / x
h = (25 - 0.5*(4.08)^2) / 4.08
h ≈ 3.77
Step 11: Calculate the volume using the values of 'x' and 'h':
V = x * (25 - 0.5x^2)
V = 4.08 * (25 - 0.5*(4.08)^2)
V ≈ 40.73 in^3
Therefore, the maximum volume of the closed box that can be made by bending the material is approximately 40.73 cubic inches.