Chapter 8.3, Problem 46AYU

a.

To determine

what angle should the pilot turn to head towards Louisville?

Answer to Problem 46AYU

  $\boxed{11.98°}$

Explanation of Solution

Given information:

In attempting to fly from Chicago to Louisville, a distance of 330 miles, a pilot inadvertently took a course that was $10°$ in error, as indicated in the figure.

If the aircraft maintains an average speed of 220 miles per hour and if the error in direction is discovered after 15 minutes, through what angle should the pilot turn to head towards Louisville?

  

Calculation:

Let us consider the following figure of revising a flight plan:

  

Let Chicago is denoted by C, Louisville is denoted by L and the Error detected point by E. So, an oblique triangle $\Delta CEL$ formed such that CL=330 miles and $\angle LCE=10°$ .

Since speed of the aircrafts is 220 miles per hour, so in 15 minutes (1/4 hour) it will cover 220/4=55 miles. It means the distance Chicago to error detected point is 55 miles.i.e., CE=55 miles. The triangle $\Delta CEL$ is an SAS triangle, using the Law of Cosines in, we get

  ${\left(EL\right)}^2={\left(CL\right)}^2+{\left(CE\right)}^2-2\left(CE\right)\left(CL\right)\cos 10°$

  $={\left(330\right)}^2+{\left(55\right)}^2-2\left(330\right)\left(55\right)\cos 10°$

  $=\text{1}0\text{89}00+\text{3}0\text{25}-\text{363}00$

  $\cos 10°$

  $=\text{76176}.\text{47857}$

  $EL\approx 276.0009miles$

Now,

  $\begin{array}{l}\cos \left(\angle CEL\right)={\frac{{\left(CE\right)}^2+{\left(EL\right)}^2-\left(CL\right)}{2\left(55\right)\left(276.0009\right)}}^2\\ \approx \frac{{\left(55\right)}^2+{\left(276.0009\right)}^2-{\left(330\right)}^2}{2\left(55\right)\left(276.0009\right)}\end{array}$

  $\approx \frac{3025+76176.49680081-108900}{30360.099}$

  $\approx -0.97821$

  $\angle CEL\approx {\cos }^{-1}\left(-0.97821\right)$

  $\approx 168.01676°$

From the figure, we have

  $\angle CEL+\mathrm{Re}quired,angle=180°$

  $\mathrm{Re}quired,angle=180°-\angle CEL$

  $\approx 180°-168.01676°$

  $\approx 11.98324°$

Hence, the pilot should turn to head towards Louisville with angle $11.98°$ approximately.

b.

To determine

What is the new average speed?

Answer to Problem 46AYU

  $\boxed{220.80072}$ mile/hour

Explanation of Solution

Given information:

In attempting to fly from Chicago to Louisville, a distance of 330 miles, a pilot inadvertently took a course that was $10°$ in error, as indicated in the figure.

What new average speed should the pilot maintain so that the total time of the trip is 90 minutes?

Calculation:

The distance have to cover from the error detected point E is 276.0009miles; that is $EL\approx 276.0009miles$ .

Since he has travelled 15 minutes. So the remaining time is

  $90-15=75$ minutes.

Hence, the required speed

  $=\frac{remaining,dis\tan ce}{remaining,time}$

  $\approx \frac{276.0009miles}{75\min utes}$

  $\approx 3.680012mile/\min ute$

Hence, the required speed is $\approx \boxed{220.80072}$ mile/hour.