128 CHAPTER 2 Functions and Their Graphs Figure 72 E N 400 mph d Plane 250 mph Plane 400 miles 600 miles Figure73 500 −50 0 2 (a) (b) TIP As in Examples 1 and 2, it is usually helpful to begin by drawing a diagram to represent the problem visually. j Now Work PROBLEM 19 Close Call? Suppose two planes flying at the same altitude are headed toward each other. One plane is flying due south at a groundspeed of 400 miles per hour and is 600 miles from the potential intersection point of the planes. The other plane is flying due west with a groundspeed of 250 miles per hour and is 400 miles from the potential intersection point of the planes. See Figure 72. (a) Build a model that expresses the distance d between the planes as a function of time t. (b) Use a graphing utility to graph d d t . ( ) = How close do the planes come to each other? At what time are the planes closest? EXAMPLE 3 Solution (a) Refer to Figure 72. The distance d between the two planes is the hypotenuse of a right triangle. At any time t, the length of the north/south leg of the triangle is t 600 400 . − At any time t, the length of the east/west leg of the triangle is t 400 250 . − Use the Pythagorean Theorem to find that the square of the distance between the two planes is d t t 600 400 400 250 2 2 2 ( ) ( ) = − + − Therefore, the distance between the two planes as a function of time is given by the model d t t t 600 400 400 250 2 2 ( ) ( ) ( ) = − + − (b) Figure 73(a) shows the graph of d d t( ) = on a TI-84 Plus CE. Using MINIMUM, the minimum distance between the planes is 21.20 miles, and the time at which the planes are closest is after 1.53 hours, each rounded to two decimal places. See Figure 73(b). Applications and Extensions 2.6 Assess Your Understanding 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure 1. Let P x y , ( ) = be a point on the graph of y x 8. 2 = − (a) Express the distance d from P to the origin as a function of x. (b) What is d if x 0? = (c) What is d if x 1? = (d) Use a graphing utility to graph d d x . ( ) = (e) For what values of x is d smallest? 2. Let P x y , ( ) = be a point on the graph of y x 8. 2 = − (a) Express the distance d from P to the point 0, 1 ( ) − as a function of x. (b) What is d if x 0? = (c) What is d if x 1? = − (d) Use a graphing utility to graph d d x . ( ) = (e) For what values of x is d smallest?
RkJQdWJsaXNoZXIy NjM5ODQ=