Chapter 7.3, Problem 105AYU

Holding Pattern An airplane is asked to slay within a holding pattern near Chicago’s O'Hare International Airport. The function $d\left(x\right)=70\sin \left(0.65x\right)+150$ represents the distance $d$, in miles, of the airplane from the airport at lime $x$, in minutes.

a. When the plane enters the holding pattern, $x=0$, how far is it from Ο’Hare?

b. During the first 20 minutes after the plane enters the holding pattern, at what time $x$ is the plane exactly 100 miles from the airport?

c. During the first 20 minutes after the plane enters the holding pattern, at what time $x$ is the plane more than 100 miles from the airport?

d. While the plane is in the holding pattern, will it ever be within 70 miles of the airport? Why?

To determine

To find:

a. When the plane enters the holding pattern, $x=0$, how far is it from O’Hare?

Answer to Problem 105AYU

a. 150

Explanation of Solution

Given:

Holding Pattern An airplane is asked to stay within a holding pattern near Chicago’s O’Hare International Airport.

The function $d\left(x\right)=70\sin \left(0.65x\right)+150$ represents the distance $d$, in miles, of the airplane from the airport at time $x$, in minutes.

Calculation:

a. When the plane enters the holding pattern, $x=0$, how far is it from O’Hare?

$d\left(x\right)=70\sin \left(0.65x\right)+150$

$d\left(x\right)=70\sin \left(0\right)+150$

$d\left(x\right)=150$

To determine

To find:

b. During the first 20 minutes after the plane enters the holding pattern, at what time $x$ is the plane exactly 100 miles from the airport?

Answer to Problem 105AYU

b. $6.06s$, $8.44s$, $15.72s$, $18.11s$

Explanation of Solution

Given:

Holding Pattern An airplane is asked to stay within a holding pattern near Chicago’s O’Hare International Airport.

The function $d\left(x\right)=70\sin \left(0.65x\right)+150$ represents the distance $d$, in miles, of the airplane from the airport at time $x$, in minutes.

Calculation:

b. During the first 20 minutes after the plane enters the holding pattern, at what time $x$ is the plane exactly 100 miles from the airport?

$d\left(x\right)=70\sin \left(0.65x\right)+150$

$100=70\sin \left(0.65x\right)+150$

$-50=70\sin \left(0.65x\right)$

$\sin \left(0.65x\right)=-\frac{5}{7}$

$0.65x={\sin }^{-1}\left(-\frac{5}{7}\right)$

$0.65x=0.7956+n\pi ,n=1,2,3$

$x=\frac{0.7956+n\pi }{0.65}$

When $n=1$, $x=6.06s$

When $n=2$, $x=8.44s$

When $n=3$, $x=15.72s$

When $n=4$, $x=18.11s$

To determine

To find:

c. During the first 20 minutes after the plane enters the holding pattern, at what time $x$ is the plane more than 100 miles from the airport?

Answer to Problem 105AYU

c. Before $6.06s$, between $8.44s$ and $15.72s$ and after $18.11s$

Explanation of Solution

Given:

Holding Pattern An airplane is asked to stay within a holding pattern near Chicago’s O’Hare International Airport.

The function $d\left(x\right)=70\sin \left(0.65x\right)+150$ represents the distance $d$, in miles, of the airplane from the airport at time $x$, in minutes.

Calculation:

c. During the first 20 minutes after the plane enters the holding pattern, at what time $x$ is the plane more than 100 miles from the airport?

Before $6.06s$, between $8.44s$ and $15.72s$ and after $18.11s$.

To determine

To find:

d. While the plane is in the holding pattern, will it ever be within 70 miles of the airport? Why?

Answer to Problem 105AYU

d. No.

Explanation of Solution

Given:

Holding Pattern An airplane is asked to stay within a holding pattern near Chicago’s O’Hare International Airport.

The function $d\left(x\right)=70\sin \left(0.65x\right)+150$ represents the distance $d$, in miles, of the airplane from the airport at time $x$, in minutes.

Calculation:

d. While the plane is in the holding pattern, will it ever be within 70 miles of the airport? Why?

No