Chapter 3, Problem 47CR

a.

To determine

The domain of the function.

Answer to Problem 47CR

The domain of the function is $\left[0,18,000\right)$ .

Explanation of Solution

Given information:

The time $t$ (in minutes) for a small plane to climb to an altitude of h feet is modeled by $t=50{\log }_{10}\left[18,000/\left(18,000-h\right)\right]$ .

Calculation:

The given function is $t=50{\log }_{10}\left[18,000/\left(18,000-h\right)\right]$

The function is defined when,

  $\begin{array}{l}\frac{18,000}{18,000-h}>0\\ 18,000-h>0\\ 18,000>h\end{array}$

And, since, height cannot be less than 0 that is negative. Therefore, the domain of the function is $\left[0,18,000\right)$ .

Hence,

The domain of the function is $\left[0,18,000\right)$ .

b.

To determine

To identify: Any asymptote and graph the function $t=50{\log }_{10}\left[18,000/\left(18,000-h\right)\right]$ .

Answer to Problem 47CR

The vertical asymptote is at $h=18,000$ .

Explanation of Solution

Given information:

The time $t$ (in minutes) for a small plane to climb to an altitude of h feet is modeled by $t=50{\log }_{10}\left[18,000/\left(18,000-h\right)\right]$ .

Calculation:

The graph of the function is drawn below:

  

From the graph the asymptotes is the vertical asymptote at $h=18,000$ .

Hence,

The vertical asymptote is at $h=18,000$ .

c.

To determine

To find: The time required to further increase its altitude when the plane approaches its absolute ceiling.

Answer to Problem 47CR

As the plane approaches its ceiling, the time required to further increase its altitude further increases.

Explanation of Solution

Given information:

The time $t$ (in minutes) for a small plane to climb to an altitude of h feet is modeled by $t=50{\log }_{10}\left[18,000/\left(18,000-h\right)\right]$ .

Calculation:

For the function $t=50{\log }_{10}\left[18,000/\left(18,000-h\right)\right]$

As the value of h approaches its ceiling that is 18000. The value $\frac{18,000}{18,000-h}$ become very large and hence t becomes very large.

Hence,

As the plane approaches its ceiling, the time required to further increase its altitude further increases.

d.

To determine

To find: The amount of time it takes for the plane to climb to an altitude of 4000 feet.

Answer to Problem 47CR

Time required is 5.46 minutes.

Explanation of Solution

Given information:

The time $t$ (in minutes) for a small plane to climb to an altitude of h feet is modeled by $t=50{\log }_{10}\left[18,000/\left(18,000-h\right)\right]$ .

Calculation:

To find the time required for the plane to climb to an altitude of 4000 feet.

Substitute $h=4000$ in the given equation and solve for t:

  $\begin{array}{l}t=50{\log }_{10}\left[\frac{18,000}{\left(18,000-h\right)}\right]\\ t=50{\log }_{10}\left[\frac{18,000}{\left(18,000-4000\right)}\right]\\ t=50{\log }_{10}\left[\frac{18,000}{14000}\right]\\ t=50{\log }_{10}\left[\frac{9}{7}\right]\\ t=5.46\end{array}$

Hence,

Time required is 5.46 minutes.