Maximizing Volume: The Box Example

Transcription

1 Maximizing Volume: The Box Example Names: Date: About this Laboratory You will construct a box from a sheet of paper with dimensions 8.5 inches x 11 inches by cutting congruent squares from each corner of the paper and folding up the sides. You will then explore the changes of the shape and volume of the box with respect to the size of the square which is cut out of each corner. You will find a function that relates the volume of the box to the size of the square. You will approximate the maximum volume and the size of square that will allow you to maximize the volume. ****************************************************************************** By cutting out congruent squares from the corners of the paper construct the open box as described above. (The squares will be discarded.) We will fill the box with rice and want to maximize the volume of the box. You will turn the box in for some points for the lab so put your name on the box. Each person should turn in a box. Let x be the cut length of the square. Figure 1 Figure 2 WV Geometry Project: Maximizing Volume - Pyzdrowski 1/08-1

2 Find the equation for the volume in terms of the height, x, of your box. Notice that x also represents the length of the side of the square that you cut out. Because x is the independent variable, you should only have numbers and the variable x in the right hand side of the equation. V = Examine your equation and then circle the type of graph that represents the equation. Line Parabola Cubic Circle Why did you choose your selection of graph type? Enter 11 and 8.5 respectively, into each of the white rectangles in the applet next to the word graph, and then choose Set. Point to a side inside the 2-D paper representation in the applet and then hold down the left mouse button to drag and form a new sized box. Release the button and choose Record. Repeat this process until you have at least 10 entries in the applet table. Explain how the volume is determined by using the numbers in the table. WV Geometry Project: Maximizing Volume - Pyzdrowski 1/08-2

3 Once you have at least 10 entries, copy your values into the given table. Notice that the model should provide two unique entries with a volume that is equal to zero. If you did not find them using the applet, look at your equation to help determine those heights. Height Volume 0 0 In the applet, select Graph and then Equat. Then in the rectangular box next to Y1, enter the right hand side of your volume equation (in terms of x). Once your equation is complete press the ENTER key and then select OK. HINT: You MUST use the * symbol for all multiplication. For instance, 2(3x + 5) must Sketch of Graph be entered as 2*(3*x + 5). Sketch a complete graph. Such a sketch should include intercepts, turning points, and end behavior. Label the axes with the appropriate terms. (Use words such as length, width, area, volume,... ). You may need to Select the Out, Xin, Xout, Yin, or Yout buttons once or twice to get a better view. Remember that the home button is a good way to get back to the default settings if things get messed up. WV Geometry Project: Maximizing Volume - Pyzdrowski 1/08-3

4 What does the x variable represent in this example? What does the y variable, or V, represent in this example? What set of numbers can be used for the x values in the equation? This is called the domain of the function. What set of numbers can be used for the y, or V, values in the equation? This is called the range of the function. What x values make sense for the problem? (Think about the values that you collected in your table from the applet.) What y values make sense for the problem? (Select Trace and then the radio button next to Trace y1. You can use the arrows in the box of the applet, or point to and while holding down the left mouse button move the trace point to help approximate an answer. You can come back later to verify your results. Think about the values that you collected in your table.) Zoom in graphically to approximate each x-intercept to within.01 and then come back later to confirm your results. Directions to find the point of intersection To graphically approximate the coordinates of a point, it may be necessary to adjust the viewing window. The following is similar to making a box using a graphics calculator. 1. Move the arrow inside the graph window. 2. Position the arrow to an area slightly above and to the left of the point. 3. Press and hold the left mouse button and drag the zoom frame to an area slightly below and to the right of the point. Release the mouse button. 4. The graph window should now display the selected region. 5. Continue to zoom in until an acceptable solution is obtained. In our case, we want the Xscale: and Yscale: located on the bottom toolbar of the applet to be less than.01 To read the coordinates of the point: 1. Point to and Select the point of intersection. 2. Record the displayed coordinates which appear at the bottom left of the Grapher. x-intercept : (, 0) Zooming Back Out: WV Geometry Project: Maximizing Volume - Pyzdrowski 1/08-4

5 1. Select the Back button to retrace your steps one at a time. 2. You may also select the Home button on the Grapher to return to the computer s default initial setting. Use the same technique two more times to approximate the other two intercepts. x-intercept : (, 0) x-intercept : (, 0) What do the values of the coordinates of the x-intercepts mean in this real life problem? (To answer the question, THINK: What is so important about these coordinates? What do they mean in terms of your units?) Now move the cursor to the local maximum or turning point on your screen. Zoom in graphically to approximate the coordinates of the point to within.01. (, ) WV Geometry Project: Maximizing Volume - Pyzdrowski 1/08-5

6 Describe the real life meaning of the local maximum coordinates with respect to the box problem. (THINK: What do the values of the coordinates mean?) Is there an absolute or largest value for V on the "whole" graph? Explain. Do we need to know a largest value on the whole graph, if it exists, to find the maximum volume? Explain. WV Geometry Project: Maximizing Volume - Pyzdrowski 1/08-6

7 Fill in the table using your equation for V. You may use the Table button in the applet to assist you. For example, to find V(0): Select Table. Fill in 0, and then select Enter. x V(x) Do Later: Explore the values of x and V(x) in the table and explain how these justify the maximum volume and restricted x values found earlier. (Make sure that you discuss how the table justifies your previous answers.) What should the cut length of the square be to maximize volume? (Be sure to use the appropriate units in your answer.) What is the maximum volume of the box? (Be sure to use the appropriate units in your answer.) WV Geometry Project: Maximizing Volume - Pyzdrowski 1/08-7