An open box is to be made out of a 8-inch by 14-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

Dimensions of the bottom of the box:

Height of the box:

see related problems at the links below. Same technique; just change the numbers.

5.6

an open box of rectangular base is to be made from 24 cm by 45cm cardboard by cutting out squares sheets of equal size from each corner and bending the sides.find the dimensions of the corner squares to obtain a box having largest volume.

an open box with dimensions 2 by 3 by 4 inchesneeds to be increased in size to hold 5 times as much material as the current box. ehat would be the new dimension of the new box?

To find the dimensions of the resulting box that has the largest volume, we need to optimize the volume function with respect to the dimensions of the bottom of the box.

Let's define the length of the square to be cut out from each corner as "x". Since the cardboard piece is initially 8 inches by 14 inches, the length of the resulting bottom of the box will be 8 - 2x inches, and the width will be 14 - 2x inches.

The height of the resulting box will be the same as the length of the square cut out from the corners, which is "x" inches.

Now we can calculate the volume of the box using these dimensions. The volume (V) of a rectangular box is given by V = length * width * height.

Therefore, the volume (V) of the box will be V = (8 - 2x) * (14 - 2x) * x.

To find the dimensions of the resulting box that has the largest volume, we need to find the value of "x" that maximizes the volume function V(x).

To do this, we can take the derivative of V(x) with respect to "x", set it equal to zero, and solve for "x".

dV/dx = 0

Taking the derivative of V(x) with respect to "x" gives us:

dV/dx = (14 - 2x) * (8 - 2x) + (8 - 2x) * (14 - 2x) + x * (-2)

Simplifying this expression gives us:

0 = -4x^2 + 36x - 224

Now we can solve this quadratic equation to find the value(s) of "x" that make the derivative equal to zero:

-4x^2 + 36x - 224 = 0

To solve this quadratic equation, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.

In this case, a = -4, b = 36, and c = -224.

Using the quadratic formula, we get two values for "x":

x = (-36 ± √(36^2 - 4(-4)(-224))) / (2(-4))

Solving this equation gives two values for "x": x = 3.5 and x = 8.

However, since we are cutting squares out from the corners, the value of "x" cannot be greater than half the length or width of the cardboard piece. In this case, half of 8 inches or 14 inches is 4 inches.

Therefore, we can only consider the value of "x" that is less than or equal to 4 inches. Therefore, x = 3.5 inches.

So, the dimensions of the bottom of the box will be 8 - 2x = 8 - 2(3.5) = 8 - 7 = 1 inch and 14 - 2x = 14 - 2(3.5) = 14 - 7 = 7 inches.

Finally, the height of the box will be x = 3.5 inches.

Therefore, the dimensions of the resulting box that has the largest volume are a bottom of 1 inch by 7 inches and a height of 3.5 inches.