Chapter 4.8, Problem 32E

a.

To determine

Find the plane’s altitude after $1$ minute.

Answer to Problem 32E

  $\boxed{5098.78}$ feet.

Explanation of Solution

Given information:

During takeoff, an airplane’s angle of ascent is $180°$ and its speed is $260$  feet per second.

Find the plane’s altitude after $1$ minute.

Calculation:

Consider the following figure which describes the path of path of the airplane travelling at the speed of $275$ feet per second and having an angle of ascent as $18°$

  

Since the distance is given by the product of the speed and time, therefore the distance travelled by the airplane in $1$ minute is

  $\begin{array}{l}s=vt\\ =275\times 60\\ =16500\end{array}$

Note that the above triangle is a right angle triangle where hypotenuse represents the distance $s$

Since sine of the angle is given by the ratio of the length of the perpendicular $o$ the hypotenuse, therefore

  $\begin{array}{l}8\sin \left(18°\right)=\frac{h}{s}\\ h=s\sin \left(18°\right)\\ =16500\times \sin \left(18°\right)\\ =5098.78\end{array}$

Therefore, the plane’s altitude after $1$ minute is $\boxed{5098.78}$ feet

b

To determine

How long will it take for the plane to climb to an altitude of $10,000$ feet?

Answer to Problem 32E

  $\boxed{1.96\min utes}$

Explanation of Solution

Given information:

During takeoff, an airplane’s angle of ascent is $180°$ and its speed is $260$  feet per second.

How long will it take for the plane to climb to an altitude of $10,000$ feet?

Calculation:

Since sine of the angle is given by the ratio of the length of the perpendicular $o$ the hypotenuse, therefore when $h=10,000$ ft

  $\begin{array}{l}\sin \left(18°\right)=\frac{h}{s}\\ s=\frac{h}{\sin \left(18°\right)}\\ =\frac{10000}{\sin \left(18°\right)}\\ =32360.68\end{array}$

Since the distance is given by the product of the speed and time, therefore the time taken for the plane to climb to an altitude of $10,000$ feet is

  $\begin{array}{l}t=\frac{32360.68}{275}\\ =117.68\end{array}$

Thus, the time taken to climb to an altitude of $h=10,000$ ft is $117.68$ seconds, that is $\boxed{1.96\min utes}$ .