A square sheet of cardboard with a side 16 inches is used to make an open box by cutting squares of equal size from the four corners and folding up the sides. What size squares should be cut from the corners to obtain a box with largest possible volume?
v = x(16-2x)^2 = 2x(8-x)^2
dv/dx = 4(3x-8)(x-8)
x=8 is obviously a minimum (v=0), so
max volume is at x = 8/3
To find the size of squares that should be cut from the corners to obtain a box with the largest possible volume, follow these steps:
Step 1: Visualize the problem
Visualize the square sheet of cardboard with a side length of 16 inches.
Step 2: Identify the variables
Let x represent the size of the squares to be cut from each corner.
Step 3: Determine the dimensions of the box
When the squares are cut and the sides are folded up, the resulting box will have dimensions of (16-2x) inches by (16-2x) inches by x inches.
Step 4: Calculate the volume of the box
The volume V of the box is given by multiplying the dimensions: V = (16-2x)*(16-2x)*x.
Step 5: Simplify the volume equation
Simplify the volume equation by multiplying the factors and combining like terms: V = 4x^3 - 64x^2 + 256x.
Step 6: Find the maximum volume
To find the maximum volume, we need to find the value of x that maximizes the volume. This can be done by finding the critical points of the function.
Step 7: Take the derivative
Take the derivative of the volume function with respect to x: V' = 12x^2 - 128x + 256.
Step 8: Set the derivative equal to zero
Set the derivative equal to zero and solve for x: 12x^2 - 128x + 256 = 0.
Step 9: Solve for x
To solve this quadratic equation, you can use the quadratic formula: x = [-(-128) ± sqrt((-128)^2 - 4(12)(256))] / (2(12)).
Simplify the equation to find the two possible values of x.
Step 10: Evaluate the critical points
Evaluate the critical points to determine which value of x gives the maximum volume. Substitute the values into the volume equation.
Step 11: Find the largest possible volume
Compare the volumes obtained from the critical points and choose the largest value of V.
Therefore, the size of the squares that should be cut from the corners to obtain a box with the largest possible volume is the value of x found in Step 11.
To find the size of the squares that should be cut from the corners to obtain a box with the largest possible volume, we need to consider the following steps:
Step 1: Visualize the Problem
Imagine the square sheet of cardboard and visualize cutting equal squares from its corners. Then fold up the resulting flaps to create an open-top box. The size of the squares cut from the corners will determine the height and dimensions of the box.
Step 2: Define the Variables
Let's assume that the size of the squares cut from each corner is represented by "x" inches. This means the length and width of the base of the box will be reduced by 2x inches (2x from each side), and the height of the box will be x inches.
Step 3: Determine the Expressions
From the given information, we can determine the dimensions of the open-top box:
- The length of the base: 16 inches - (2 * x) inches = 16 - 2x inches
- The width of the base: 16 inches - (2 * x) inches = 16 - 2x inches
- The height of the box: x inches
Step 4: Calculate the Volume
The volume of the open-top box can be calculated by multiplying the length, width, and height:
Volume = (16 - 2x) * (16 - 2x) * x
= (256 - 64x + 4x^2) * x
= 4x^3 - 64x^2 + 256x
Step 5: Find the Maximum Volume
To find the maximum volume, we'll take the derivative of the volume with respect to "x" and set it equal to zero, then solve for "x". The resulting value of "x" will give us the size of the squares to be cut from the corners, leading to the box with the maximum volume.
d/dx(4x^3 - 64x^2 + 256x) = 0
12x^2 - 128x + 256 = 0
Solving for "x" gives us x = 4.
Therefore, cutting squares of size 4 inches from each corner will result in a box with the largest possible volume.