A manufacturer constructs open boxes from sheets of cardboard that are 6 inches square by cutting small squares from the corners and folding up the sides. The Research and Development Department asks you to determine the size of the square that produces a box of greatest volume. Proceed as follows. Let x be the length of a side of the square to be cut and V be the volume of the resulting box. Show that V = x(6-2x)^2 (sketched it already.)
Are there any restrictions on the value of x? Explain.
Estimate the largest volume.
This is a continuation of the question I answered for you earlier.
Since you titled it "Calculus" you should be able to finish it.
1. expand and simplify the expression for V
2. differentiate, you will have a quadratic
3. solve that quadratic by setting it equal to zero for a max of V
4. sub the value you found in 3. into the original volume equation
for the restriction, all you have to do is look at the equation for V
V of course has to be positive.
so x > 0 and x < 3 or else the base side values make no sense.
restriction on x : 0 < x < 3
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To determine any restrictions on the value of x, we consider the construction process.
The manufacturer constructs open boxes by cutting small squares from the corners of a larger square sheet. The resulting box has dimensions given by (6-2x), where (6-2x) represents the length and width of the base and x represents the height (or depth) of the box.
For a valid box to be constructed, the dimensions must be positive, meaning that (6-2x) > 0 and x > 0. Solving the inequality (6-2x) > 0, we find:
6 - 2x > 0
-2x > -6
x < 3
So, the value of x must be less than 3 for a valid box to be constructed. This is the restriction on the value of x.
To estimate the largest volume, we need to find the maximum value of the volume function V = x(6-2x)^2.
Taking the derivative of V with respect to x:
dV/dx = (6-2x)^2 + x * 2(6-2x)(-2)
Setting dV/dx to zero to find the critical point:
(6-2x)^2 + x * 2(6-2x)(-2) = 0
Simplifying and solving for x:
(6-2x)(6-2x) - 2x(6-2x) = 0
(6-2x)[(6-2x)-2x] = 0
(6-2x)(6-4x) = 0
Setting each factor to zero and solving for x:
6-2x = 0 or 6-4x = 0
2x = 6 or 4x = 6
x = 3 or x = 6/4 = 3/2 = 1.5
However, we need to consider the restriction x < 3. So, the critical point x = 3 is not valid.
Therefore, the only critical point satisfying the restriction is x = 1.5.
To determine if it is a maximum or minimum, we can take the second derivative of V.
Taking the second derivative:
d^2V/dx^2 = 2(6-2x)(-2) + 2(-2)
At the critical point x = 1.5:
d^2V/dx^2 = 2(6-2(1.5))(-2) + 2(-2) = 2(3)(-2) + 2(-2) = -12 + (-4) = -16
Since the second derivative is negative, the critical point x = 1.5 corresponds to a maximum. Hence, the largest volume is obtained when x = 1.5.
To estimate the largest volume, substitute x = 1.5 into the volume function V:
V = 1.5(6-2(1.5))^2
V = 1.5(6-3)^2
V = 1.5(3)^2
V = 1.5(9)
V = 13.5
Therefore, the largest volume is estimated to be 13.5 cubic inches when x = 1.5 inches.
To determine any restrictions on the value of x, we need to consider the dimensions of the box.
First, let's visualize the construction of the box: We start with a sheet of cardboard that is 6 inches square. We then cut squares with side length x from each of the four corners and fold up the sides to form the box.
The base of the box will have dimensions 6 - 2x (since we've removed squares of size x from each corner), and the height of the box will be x.
To find the volume of the box, we multiply the length, width, and height:
V = (6 - 2x)(6 - 2x)(x)
= x(6 - 2x)^2
Now, let's analyze the restrictions on x:
1. The length of the square to be cut cannot exceed half the side length of the original cardboard (6 inches in this case). So, we have the inequality:
x ≤ 6/2
x ≤ 3
2. The length of the square to be cut cannot be negative or zero, as that would result in either not cutting from the corners or not having a box. Therefore, we have the inequality:
x > 0
Combining these restrictions, we find that the possible values for x are:
0 < x ≤ 3
To estimate the largest volume, we can consider the behavior of the volume function V = x(6 - 2x)^2 over the interval 0 < x ≤ 3.
By examining the graph, or using calculus, we can find that the volume is maximized when x = 1.5 inches. This means that cutting squares with side length 1.5 inches from each corner will result in the box with the largest volume.