Chapter 4.8, Problem 31E

a.

To determine

Find an equation for the length $\text{l}$ of the tether you are holding in terms of $h$ .

Answer to Problem 31E

  $l=\boxed{\sqrt{h^2+34h+10289}}$

Explanation of Solution

Given information:

You are holding one of the tethers attached to the top of a giant character balloon that is floating approximately $\text{2}0\text{ feet}$ above ground level. You are standing approximately $\text{1}00\text{ feet}$ ahead of the balloon.

Find an equation for the length $\text{l}$ of the tether you are holding in terms of $h$ , the height of the balloon from top to bottom.

Calculation:

We shall use the fact that the height of the giant balloon from top to bottoms is $h$ , and its bottom is floating $20ft$ above the ground level. Also, the length of the tether attached to the top the balloon is $l$ . A person of height $3ft$ holding that tether is standing $100ft$ ahead of the balloon.

Let, the angle of elevation from the person to the top of the balloon be $\theta$ .

From the figure in right $\Delta ABC$ , distance of the balloon from the person’s head $AB=h+17$

Length $AB=h+17$

Now, in right $\Delta ABC$ , by using the Pythagoras theorem

  $l^2={\left(h+17\right)}^2+{100}^2$

  $l=\sqrt{{\left(h+17\right)}^2+{100}^2}$

  $l=\sqrt{h^2+34h+289+10000}$

Hence, $l=\boxed{\sqrt{h^2+34h+10289}}$

b.

To determine

Find an equation for the angle of elevation θ from you to the top of the balloon.

Answer to Problem 31E

  $\boxed{\theta =Arc\cos \left(\frac{100}{l}\right)}$

Explanation of Solution

Given information:

You are holding one of the tethers attached to the top of a giant character balloon that is floating approximately $\text{2}0\text{ feet}$ above ground level. You are standing approximately $\text{1}00\text{ feet}$ ahead of the balloon.

Find an equation for the angle of elevation θ from you to the top of the balloon.

Calculation:

In the right $\Delta ABC$ we can see that

  $\cos \theta =\frac{Adjacent}{\text{Hypotenuse}}$

  $=\frac{100}{l}$

  $\Rightarrow \theta ={\cos }^{-1}\left(\frac{100}{l}\right)$

Hence, $\boxed{\theta =Arc\cos \left(\frac{100}{l}\right)}$

c.

To determine

Find the height h of the balloon.

Answer to Problem 31E

  $\boxed{h=53.02ft}$

Explanation of Solution

Given information:

You are holding one of the tethers attached to the top of a giant character balloon that is floating approximately $\text{2}0\text{ feet}$ above ground level. You are standing approximately $\text{1}00\text{ feet}$ ahead of the balloon.

The angle of elevation to the top of the balloon is $\text{35}°$ . Find the height h of the balloon.

Calculation:

We shall use $\theta \text{=35}°$ . Hence in the right $\Delta ABC$ , it follows the fact

  $\tan \text{35}°=\frac{Adjacent}{\text{Opposite}}$

  $=\frac{h+17}{100}$

  $100\tan 35°=h+17$

  $0.70020\times 100=h+17$

  $70.02-17=h$

Hence, $\boxed{h=53.02ft}$