An open box is to be made from 8 inches by 18 inches piece of cardboard by cutting
squares of equal size from the four corners and bending up the sides. How long should
the sides of the squares be to obtain a box with the largest volume?
If the squares have side x, then the volume is
v = x(8-2x)(18-2x) = 4x^3 - 52x^2 + 144x
dv/dx = 12x^2-104x+144 = 4(3x^2-26x+36)
so find where dv/dx = 0 and make sure that is a maximum, not a minimum.
worksheet 4 yan ha, send answers naman dyan
To find the length of the sides of the squares to obtain the box with the largest volume, we can follow these steps:
Step 1: Let's assume that the side length of the square to be cut is "x" inches.
Step 2: After cutting squares of side length "x" from each corner, the dimensions of the resulting box will be (18-2x) inches by (8-2x) inches by "x" inches (height).
Step 3: The volume V of the box can be calculated by multiplying these three dimensions: V = (18-2x)(8-2x)x.
Step 4: Expand the equation by multiplying the terms: V = 4x^3 - 52x^2 + 144x.
Step 5: To find the maximum volume, we can take the derivative of the volume equation with respect to x and set it to zero.
dV/dx = 12x^2 - 104x + 144 = 0.
Step 6: Solve the equation for x by factoring or using the quadratic formula.
Step 7: Once you have the value of x, substitute it back into the volume equation to find the maximum volume.
Step 8: Compare the obtained maximum volume with the volume of other values of x to ensure it is indeed the maximum volume.
Step 9: The side length of the squares should be the value of x that gives the maximum volume.
To find the length of the sides of the squares that will result in the largest volume of the box, we can use optimization techniques. Here's how you can solve it step by step:
1. Let's assume that the side length of the square to be cut from the corners is 'x' inches.
2. Now, we need to determine the dimensions of the box after cutting and folding the cardboard.
- The length of the box will be (18 - 2x) inches, as we are removing a square of side length 'x' from both ends.
- The width of the box will be (8 - 2x) inches for the same reason.
- The height of the box will be 'x' inches, as the squares we cut will be folded to form the sides.
3. The volume of the box can be calculated by multiplying the length, width, and height:
Volume = length * width * height = (18 - 2x) * (8 - 2x) * x.
4. Now, we need to find the maximum value of this volume function.
To find the value of 'x' that maximizes the volume, we can differentiate the volume function with respect to 'x' and find when the derivative equals zero. Let's do that:
1. Differentiate the volume function:
dV/dx = (18 - 2x)(8 - 2x)x' + (18 - 2x)x(8 - 2x)' + (8 - 2x)(18 - 2x)x'.
2. Simplify:
dV/dx = (18 - 2x)(8 - 2x)(1) + (18 - 2x)(-2)(8 - 2x) + (8 - 2x)(-2)(18 - 2x).
3. Expand and simplify further:
dV/dx = 4x^3 - 52x^2 + 144x.
Now, set this derivative equal to zero and solve for 'x':
4x^3 - 52x^2 + 144x = 0.
We can factor out '4x' from the equation:
4x(x^2 - 13x + 36) = 0.
This equation will be zero when either '4x' or '(x^2 - 13x + 36)' equals zero. We set each factor equal to zero and solve for 'x' separately:
a) 4x = 0.
We get x = 0.
b) (x^2 - 13x + 36) = 0.
We can't factorize this equation further, so we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a),
where a = 1, b = -13, and c = 36.
Applying the values, we get:
x = (13 ± √(169 - 144)) / 2,
x = (13 ± √25) / 2,
x = (13 ± 5) / 2.
Therefore, x can be 4 or 9.
Now we have three possible values for 'x': 0, 4, and 9.
Since the purpose is to cut squares from the corners, we cannot have a side length of 0. Therefore, we need to evaluate the volume for side lengths of 4 and 9 and determine which one gives the maximum volume.
For x = 4:
Volume = (18 - 2x)(8 - 2x)x = (18 - 2(4))(8 - 2(4))(4) = 10 * 0 * 4 = 0.
For x = 9:
Volume = (18 - 2x)(8 - 2x)x = (18 - 2(9))(8 - 2(9))(9) = 0 * (-10) * 9 = 0.
Both x = 4 and x = 9 give a volume of 0, which means that the largest volume is obtained when x = 0.
Therefore, the box with the largest volume cannot be created by cutting squares from the piece of cardboard.