Maximum Volume of the Box
- Jay Lietzow
You are provided a 10 cm by 10 cm sheet of metal and asked to make it into a box by cutting out 4 corners and folding the sides up. How can you make the box with the maximum volume?
1) Drag the slider for c to see the problem illustration. Note that as you increase the value of the slider, the size of the cutout of each corner increases. As you increase the size of the cutout, c, what happens to the area of the base of the box? 2) Can you define a function for the sides of the base of the box in terms of c? Once you have noted your answer, proceed to step 2 by clicking the arrow at the bottom of the screen. 3) Now that you have defined the dimensions of the base of the box, how would you calculate the area? How would you calculate the volume of the box? Once you have noted your answer, proceed to step 3 by clicking the arrow again. 4) Do you have a guess at about what size you should cut to maximize the volume? Click ahead to step 4 to see the values for the volume computed as you move the slider. 5) Now that we have this information, we can estimate the optimal solution by trial and error (by moving the slider back and forth). What size cutout looks like it creates the box with the maximum volume? 6) How else could we find the maximum volume? Plot the function v(x) on the graph at the right by selecting that window and entering the function v(x) in the input bar. 7) Add a point the function v(x) by entering the following in the input bar: Z=(c,v(c)). Drag the slider for c back and forth and watch the point Z move on the curve. What does the y-coordinate of Z represent? 8) What value of c creates the maximum volume of the box? What is the maximum volume of the box?