An open box is to be made from a eighteen-inch by eighteen-inch square piece of material by cutting equal squares from the corners and turning up the sides (see figure). Find the volume of the largest box that can be made.
figure ( a box and one side is 18-2x)
v= ? in^3
v = x(18-2x)^2 = 4x^3-72x^2+324x
Now just find where dv/dx=0 and that will be the maximum (or minimum!) volume. I'm sure you will be able to tell which one.
so i got x=9 but this is not correct
Of course x=9 is incorrect. If you cut off a 9" corner from the 18" square, the volume is zero! Probably not the maximum value.
dv/dx=0 at another value of x. Use that one.
fight
To find the volume of the largest box that can be made, we need to determine the dimensions that will maximize the volume.
Let's assume that by cutting equal squares from each corner, the side length of each square cut is 'x'.
Now, when we fold up the sides, the resulting box will have length and width equal to (18 - 2x) inches, and the height will be 'x' inches.
Therefore, the volume of the box can be calculated by multiplying the length, width, and height:
V = (18 - 2x) * (18 - 2x) * x
To find the maximum volume, we can differentiate this volume equation with respect to 'x' and set the derivative equal to zero to find the critical points:
dV/dx = 0
Let's find the derivative of V with respect to 'x':
dV/dx = (36 - 4x) * (18 - 2x) + (18 - 2x) * (18 - 2x) - 2x * (36 - 4x)
Simplifying the equation:
dV/dx = 4x^2 - 72x + 324 + 4x^2 - 72x + 324 - 72x + 8x^2
dV/dx = 16x^2 - 216x + 648
Now, set the derivative equal to zero and solve for 'x':
16x^2 - 216x + 648 = 0
We can simplify this equation by dividing by 8:
2x^2 - 27x + 81 = 0
This quadratic equation can be factored:
(2x - 9)(x - 9) = 0
Setting each factor equal to zero:
2x - 9 = 0 or x - 9 = 0
Solving for 'x':
2x = 9 or x = 9
Since the width of the square material is 18 inches, we cannot have the side length of the square cut exceeding 9 inches.
So, the largest value for 'x' is 9 inches.
Now, substitute this value of 'x' into the volume equation:
V = (18 - 2(9)) * (18 - 2(9)) * 9
V = 0 * 0 * 9
V = 0 cubic inches
Therefore, the largest box that can be made will have a volume of zero cubic inches since the height will be zero.