Although a graphing calculator is limited to displaying the relationship between only two...

Although a graphing calculator is limited to displaying the relationship between only two variables (for the most part), it has a feature that enables us to see how these two are related with respect to a third. Consider the variation equation from Example 8 in Section 2.6: F = 0.05dv2. If we want to investigate the relationship between fuel consumption and velocity, we can have the calculator display multiple versions of the relationship simultaneously for different values of d. This is accomplished using the “{” and “}” symbols, which are functions to the parentheses. When the calculator sees values between these grouping symbols and separated by commas, it is programmed to use each value independently of the others, graphing or evaluating the relation for each value in the set. We illustrate by graphing the relationship f = 0.05dv2 for three different values of d. Enter the equation on the screen as Y1 = 0.05{10, 20, 30}X2, which tells the calculator to graph the equations Y1 = 0.05(10)X2, Y1 = 0.05(20)X2, and Y1 = 0.05(30)X2 on the same grid. Note that since d is constant, each graph is a parabola. Set the viewing window using the values given in Example 8 as a guide. The result is the graph shown in Figure 2.97, where we can study the relationship between these three variables using the up and down arrows. From our work with the toolbox functions and transformations, we know the widest parabola used the coefficient “10,” while the narrowest parabola used the coefficient “30.” As shown, the graph tells us that at a speed of 15 nautical miles per hour (X = 15), it will take 112.5 barrels of fuel to travel 10 mi (the first number in the list). After pressing the key, the cursor jumps to the second curve, which shows values of X = 15 and Y = 225. This means at 15 nautical miles per hour, it would take 225 barrels of fuel to travel 20 mi. Use these ideas to complete the following exercises:

Figure 2.97

The maximum safe load S for a wooden horizontal plank supported at both ends varies jointly with the width W of the beam, the square of its thickness T, and inversely with its length L. A plank 10 ft long, 12 in. wide, and 1 in. thick will safely support 450 lb. Find the value of k and write the variation equation, then use the equation to explore:

a. Safe load versus thickness for a constant width and given lengths (quadratic function). Use w = 8 in. and {8, 12, 16} for L.

b. Safe load versus length for a constant width and given thickness (reciprocal functional). Use w = 8 in. and for thickness.