An open box with a square base is to be made from a square piece of cardboard 24 inches on a side by cutting out a square of side x inches from each corner and turning up the sides.Graph V=V(x)

Each side of the base will be (24-2x)

v(x) = x(24-x)^2

Wolfram's magic done here:

http://www.wolframalpha.com/input/?i=plot+y+%3D+x%2824-x%29%5E2

for a better graph, substitute (24-2x)

x(24-2x)^2

Sure, here's the graph V = V(x):

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x x x

I hope that brings a smile to your face! 😄

To graph the volume V as a function of x, we need to understand how the volume of the open box changes as the size of the squares cut out and folded up vary.

Let's break down the problem step by step:

1. Start with a square piece of cardboard measuring 24 inches on each side.

2. We cut out a square piece of size x inches from each corner of the cardboard.

3. Fold up the remaining sides of the cardboard to create an open box.

To find the volume, we need to determine the dimensions of the open box and then use the formula for volume.

The length and width of the base of the box will be the side length of the original square minus twice the length of the squares cut out, so (24 - 2x).
The height of the box will be equal to the length of the squares cut out, so x.

Now, we can write the formula for the volume of the box in terms of x:
V(x) = (24 - 2x) * (24 - 2x) * x

To graph V(x), we will plot the values of V for different values of x.

Here are some steps to help you create the graph:

1. Choose a range for x. Let's say, for example, x can vary from 0 to 12 inches. You can adjust this based on the problem or the desired range.

2. Divide the x-axis into equal intervals based on the range of x you have chosen.

3. Calculate the corresponding values of V for each value of x using the formula V(x) = (24 - 2x) * (24 - 2x) * x.

4. Plot the points (x, V) on the graph, where x is the value of x for each interval and V is the corresponding value of V obtained in step 3.

5. Connect the points with a smooth curve. If the function is continuous, the curve should be smooth without any discontinuities or jumps.

Remember to label the axes of the graph with x and V to indicate which variable is being represented on each axis.

This should give you a graph of V as a function of x.