Chapter 10.7, Problem 58AYU

Uniform Motion A Cessna (heading south at $120$ mph) and a Boeing $737$ (heading west at $600$ mph) are flying toward the same point at the same altitude. The Cessna is $100$ miles from the point where the flight patterns intersect, and the $737$ is $550$ miles from this intersection point. See the figure.

Find parametric equations that model the motion of the Cessna and the $737.$

Find a formula for the distance between the planes as a function of time.

Graph the function in part (b) using the graphing utility.

What is the minimum distance between the planes? When are the planes closest?

Simulate the motion of the planes by simultaneously graphing the equations found in part (a).

To determine

The parametric equation that model of the Cessna and the Boeing $737$.

Answer to Problem 58AYU

Solution:

The parametric equation of the Cessna is $x\left(t\right)=0,y\left(t\right)=100-120t$ and the Boeing $737$ is $x\left(t\right)=550-600t,y\left(t\right)=0$.

Explanation of Solution

Given information:

A Cessna (heading south at $120$ mph) and a Boeing $737$(heading south at $600$ mph) are flying towards the same point at the same altitude. The Cessna is $100$ miles from the point where the flight patterns intersect, and the Boeing $737$ is $550$ miles from this intersection point.

Explanation:

Let the motion of Cessna be $x\left(t\right)$, and that of the Boeing $737$ be $y\left(t\right)$.

A Cessna and a Boeing $737$ are flying towards the same point at the same altitude.

So the sign of distance is negative.

Here, the $\text{distance =}\text{speed}\times \text{time}$.

The heading speed of Cessna along $x-$ direction is zero.

The heading speed of Cessna along $y-$ direction is $v=-120$ mph.

The heading speed of the Boeing $737$ along $x-$ direction is $u=-600$ mph

The heading speed of the Boeing $737$ along $y-$ directionis $0$.

The initial distance of Cessna from the point the intersection is $100$ miles, and that of the Boeing $737$ is $550$ miles.

The distance from the point of intersection is equal to the initial distance plus the product of speed and time $t$.

$\text{Distance from intersection}=\text{initial distance}\text{+ speed}\times \text{time}$

The parametric equation that models the motion of the Cessna and the Boeing $737$ is $x=x_0+ut$ and $y=y_0+vt$.

The parametric equation that models the motion of the Cessna is $x\left(t\right)=0,y\left(t\right)=100-120t$.

The parametric equation that models the motion of the Cessna is $x\left(t\right)=550-600t,y\left(t\right)=0$.

Thus, the parametric equation of the Cessna is $x\left(t\right)=0,y\left(t\right)=100-120t$, and that of the Boeing $737$ is $x\left(t\right)=550-600t,y\left(t\right)=0$.

To determine

The formula for distance between the planes as a function of time.

Answer to Problem 58AYU

Solution:

The formula for distance between the planes is $f\left(t\right)=\sqrt{{\left(100-120t\right)}^2+{\left(550-600t\right)}^2}$.

Explanation of Solution

Given information:

A Cessna (heading south at $120$ mph) and a Boeing $737$(heading south at $600$ mph) are flying towards the same point at the same altitude. The Cessna is $100$ miles from the point where the flight patterns intersect, and the Boeing $737$ is $550$ miles from this intersection point.

Explanation:

The positions of the flight form a right angle triangle.

Let $f\left(t\right)$ be the distance between the two planes; it is a function of time.

To find the distance between the two planes by using the Pythagorean Theorem, Here, $x\left(t\right)=100-120t$ and $y\left(t\right)=550-600t$.

By using the Pythagorean Theorem $\sqrt{{\left(x\right)}^2+{\left(y\right)}^2}$,

$f\left(t\right)=\sqrt{{\left(100-120t\right)}^2+{\left(550-600t\right)}^2}$

$=\sqrt{{\left(100-120t\right)}^2+{\left(550-600t\right)}^2}$

Thus, the distance between the two planes is $f\left(t\right)=\sqrt{{\left(100-120t\right)}^2+{\left(550-600t\right)}^2}$.

To determine

To graph: The function in part (b) using the graphing utility.

Explanation of Solution

Given information:

The function of time is $f\left(t\right)=\sqrt{{\left(100-120t\right)}^2+{\left(550-600t\right)}^2}$.

Graph:

Step I: Press the ON key.

Step II: Now, press [Y=]. Input the right hand side of the function $f\left(x\right)=\sqrt{{\left(100-120x\right)}^2+{\left(550-600x\right)}^2}$ in $Y_1$

Step IV: Press [WINDOW] key, and set the viewing window as below:

$\begin{array}{ll}X\min =-2\hfill & X\max =4X\text{scl}=1\hfill \\ Y\min =-50\hfill & Y\max =175Y\text{scl}=1\hfill \end{array}$ $$.

Step IV: Then hit [Graph] key to view the graph.

Interpretation:

Distance between Cessna and Boeing 737 decreases and then increases.

To determine

The minimum distance between the planes and time when the planes are closest.

Answer to Problem 58AYU

Solution:

The minimum distance between the planes is $9.806$ miles, and the planes are closest at the time $0.913$ hours.

Explanation of Solution

Given information:

A Cessna (heading south at $120$ mph) and a Boeing $737$(heading south at $600$ mph) are flying towards the same point at the same altitude. The Cessna is $100$ miles from the point where the flight patterns intersect, and the Boeing $737$ is $550$ miles from this intersection point.

Explanation:

The function for distance between the planes is $f\left(t\right)=\sqrt{{\left(100-120t\right)}^2+{\left(550-600t\right)}^2}$.

To find the minimum distance between the planes by using graphing utility, Step I: Press the ON key.

Step II: Now, press [Y=]. Input the right hand side of the function $f\left(x\right)=\sqrt{{\left(100-120x\right)}^2+{\left(550-600x\right)}^2}$ in $Y_1$

Step IV: Press [WINDOW] key, and set the viewing window as below:

$\begin{array}{ll}X\min =-2\hfill & X\max =4X\text{scl}=1\hfill \\ Y\min =-50\hfill & Y\max =175Y\text{scl}=1\hfill \end{array}$ $$.

Step IV: Then hit [Graph] key to view the graph.

Step V: Press [2nd] [TRACE] to access the calculate menu.

Step VI: Press [3] to the minimum.

Step VII: If necessary, repeatedly press the up-down arrow keys until the appropriate function appears in the border at the top of the screen.

Step VIII: Set the left bound of the minimum point.

Step IX: Set the right bound of the minimum point.

Step X: Press the Enter twice for the minimum point.

From the above graph, the minimum distance between the planes is $9.806$ miles, and the planes are closest at $0.913$ hours.

To determine

To graph: The Simulate motion of the planes by simultaneously graphing the equation found in part (a).

Explanation of Solution

Given information:

A Cessna (heading south at $120$ mph) and a Boeing $737$(heading south at $600$ mph) are flying towards the same point at the same altitude. The Cessna is $100$ miles from the point where the flight patterns intersect, and the Boeing $737$ is $550$ miles from this intersection point.

Graph:

From part (a),

Parametric equations to describe the Cessna motion and the Boeing $737$ motion are given below:

Cessna: $x\left(t\right)=0,y\left(t\right)=100-120t$

Boeing $737$: $x\left(t\right)=550-600t,y\left(t\right)=0$

From part (d),

The planes are closest at $t=0.913$ hours.

In parametric mode with ${\text{T}}_{\text{step}}=0.01$, simultaneously graph the parametric equation.

For $0\le t\le 1$.

The relative position of Cessna and Boeing $737$ for $t=0,t=0.5,t=0.913$ and $t=1$:

The relative positions of Cessna and Boeing $737$ for $t=0$ hour are shown in the graph below:

The relative positions of Cessna and Boeing $737$ for $t=0.5\text{hour}$ are shownin the graph below:

The relative positions of Cessna and Boeing $737$ for $t=0.913\text{hour}$ are shown in the graph below:

The relative positions of Cessna and Boeing $737$ for $t=1\text{hour}$ are shown in the graph below:

Interpretation:

The graphs show the relative positions of Cessna and Boeing $737$.