Chapter 3.6, Problem 24E

Work these problems. (See Example 2.)

NASA The failure of several O-rings in field joints was the cause of the fatal crash of the Challenger space shuttle. NASA data from 24 successful launches prior to Challenger suggested that O-ring failure was related to launch temperature by a function similar to

$N\left(t\right)=\frac{600-7t}{4t-100}\left(50\le t\le 85\right),$

where t is the temperature (in °F) at launch and N is the approximate number of O-rings that fail. Assume that this function accurately models the number of O-ring failures that would occur at lower launch temperatures (an assumption NASA did not make).

a. Does $N\left(t\right)$ have a vertical asymptote? At what value of t does it occur?

b. Without actually graphing the function, what would you conjecture that the graph would look like just to the right of the vertical asymptote? What does this suggest about the number of O-ring failures that might be expected near that temperature? (The temperature at the Challenger launching was 31°.)

c. Confirm your conjecture by graphing $N\left(t\right)$ between the vertical asymptote and $t=5$.