You are making a box out of an 11in by 8in piece of poster board by cutting congruent squares out of the four corners and folding up the edges. To the nearest inch, what size square should you use to maximize the volume?
To find the size of the square that will maximize the volume of the box, we need to determine the dimensions of the box when it is formed from an 11in by 8in piece of poster board with congruent squares cut out of the corners.
Let's assume we cut out a square of size x inches from each of the four corners. After cutting the squares, the poster board remaining will have dimensions (11 - 2x) inches by (8 - 2x) inches.
To form a box, we fold up the sides of the poster board. The height of the box will be the size of the square we cut out, which is x inches.
The volume of the box is then calculated by multiplying the dimensions:
Volume = (11 - 2x) * (8 - 2x) * x
To find the maximum volume, we need to find the value of x that maximizes this expression. We can do this by finding the critical points of the equation.
First, let's simplify the equation:
Volume = x(11 - 2x)(8 - 2x)
Next, let's expand and simplify:
Volume = x(88 - 22x - 16x + 4x^2)
Volume = 4x^3 - 38x^2 + 88x
To find the critical points, we take the derivative of the volume equation with respect to x:
d(Volume)/dx = 12x^2 - 76x + 88
Setting this derivative equal to zero and solving for x:
12x^2 - 76x + 88 = 0
This is a quadratic equation. We can solve it using the quadratic formula:
x = (-(-76) ± √((-76)^2 - 4 * 12 * 88)) / (2 * 12)
Simplifying this equation gives us two values for x: x ≈ 1.27 and x ≈ 4.73.
Since we are looking for the size of the square to the nearest inch, we can round these values to the nearest whole number: x = 1 and x = 5.
Next, we need to compare the volume of the boxes formed by cutting squares of size 1 inch and 5 inches.
When x = 1 inch:
Volume = 1(11 - 2 * 1)(8 - 2 * 1) = 1 * 9 * 6 = 54 cubic inches
When x = 5 inches:
Volume = 5(11 - 2 * 5)(8 - 2 * 5) = 5 * 1 * (-2) = -10 cubic inches
Since we need a positive volume, the square size of 5 inches is not viable.
Therefore, the size of the square we should use to maximize the volume, to the nearest inch, is 1 inch. The maximum volume, to the nearest cubic inch, will be 54 cubic inches.