An open box is to be made from a 11 inch by 11 inch piece of cardboad. this box is constructed by cutting squares that measure x inches on each side from the corners of the cardboard and turning up the sides. Use a graphical calculator to find the height of the box that yields the maximum volume.
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To find the height that yields the maximum volume, we need to first determine the volume of the open box in terms of x.
Let's call the height of the box h and the length of each side of the squares cut out x.
The base of the open box will have dimensions of (11 - 2x) inches by (11 - 2x) inches. The height of the box will be h inches.
Therefore, the volume of the box can be expressed as V = (11 - 2x) * (11 - 2x) * h.
To determine the value of x that maximizes the volume, we can find the critical points by taking the derivative of the volume function with respect to x and setting it equal to zero:
dV/dx = 0
Let's use a graphical calculator to find the derivative and solve for x.
To find the height of the box that yields the maximum volume, we need to first determine the equation that represents the volume of the box as a function of the side length, x.
Let's start by visualizing the scenario. We have a rectangular piece of cardboard measuring 11 inches by 11 inches. We cut squares with side length x inches from each corner and then fold up the remaining sides to form the open box.
To construct the box, we remove squares from all four corners, leaving a rectangle with a length of (11 - 2x) inches and a width of (11 - 2x) inches. The height of the box will be x inches.
The volume of the box can be calculated by multiplying the length, width, and height together:
Volume = length * width * height
= (11 - 2x) * (11 - 2x) * x
To find the maximum volume, we need to find the value of x that maximizes this function.
To do this, we can use a graphical calculator or software like Desmos to plot the volume function as a graph and determine the highest point on the graph.
Here's how to use a graphical calculator to find the height of the box that yields the maximum volume:
1. Open a graphical calculator software or website like Desmos.
2. Enter the volume function as follows: y = (11 - 2x)(11 - 2x)x
3. Adjust the viewing window to appropriate values. Since x represents the side length of the removed squares, it cannot be negative and can go up to a maximum value of 5.5 inches (half of the original side length). Set the x-axis range to [0, 5.5] and the y-axis range to [0, 100] for better visibility.
4. Plot the graph.
5. Observe the shape of the graph and look for the highest point.
6. The x-coordinate of the highest point on the graph represents the value of x that yields the maximum volume.
7. Read the corresponding y-coordinate to find the maximum volume.
8. The value of x will be the side length of the squares cut from the corners, and the corresponding y-coordinate will be the maximum volume of the box.
9. Round the values to appropriate decimal places according to the given context, if required.
Using this method, you can visually determine the height of the box (x) that yields the maximum volume.