Chapter 5.6, Problem 31E

(a)

To determine

To find: The distance from the balloon to the soccer fields.

Answer to Problem 31E

18,885 feet to the soccer fields

Explanation of Solution

Given information: A hot air balloon is flying above Groveburg. To the left side of the balloon, the balloonist measures the angle of depression to the Groveburg soccer fields to be ${20}^{\circ }15\text{'}$ . To the right side of the balloon, the balloonist measures the angle of depression to the high school football field to be ${62}^{\circ }30\text{'}$ . The distance between the two athletic complexes is 4 miles.

Calculation:

We converted the angles to decimal form by dividing the minutes by 60. We subtracted the two angles of depression from 180 to get the angle between the lines of sight from the balloon to the soccer and football fields. It is ${97.25}^{\circ },$ we will use the law of sines to find s and f in the sketch. I have also changed the 4 miles to 21,120 feet.

Since the horizontal line drawn is parallel to the ground, the angle from the football field to the balloon is an alternate interior angle with that side of the triangle and therefore equal to the angle of depression on that side ( ${62.5}^{\circ }$ and the angle from the soccer field to the balloon is an alternate interior angle with the side of the triangle on that side and equal to that angle of depression ${20.25}^{\circ }$ ).

Here is an updated sketch with only the triangle information we need.

  $\begin{array}{l}\frac{21,120}{\sin {97.25}^{\circ }}=\frac{s}{\sin {62.5}^{\circ }}=\frac{f}{\sin {20.25}^{\circ }}\\ s=\frac{21,120\times \sin {62.5}^{\circ }}{\sin {97.25}^{\circ }}=18,885\text{ feet to the soccer field}\end{array}$

Thus,18,885 feet to the soccer fields

(b)

To determine

To find: The distance from the balloon to the football field.

Answer to Problem 31E

7,369 feet to the football field

Explanation of Solution

Given information: Ballooning: A hot air balloon is flying above Groveburg. To the left side of the balloon, the balloonist measures the angle of depression to the Groveburg soccer fields to be ${20}^{\circ }15\text{'}$ . To the right side of the balloon, the balloonist measures the angle of depression to the high school football field to be ${62}^{\circ }30\text{'}$ . The distance between the two athletic complexes is 4 miles.

Calculation:

We converted the angles to decimal form by dividing the minutes by 60. We subtracted the two angles of depression from 180 to get the angle between the lines of sight from the balloon to the soccer and football fields. It is ${97.25}^{\circ },$ we will use the law of sines to find s and f in the sketch. I have also changed the 4 miles to 21,120 feet.

Since the horizontal line drawn is parallel to the ground, the angle from the football field to the balloon is an alternate interior angle with that side of the triangle and therefore equal to the angle of depression on that side ( ${62.5}^{\circ }$ and the angle from the soccer field to the balloon is an alternate interior angle with the side of the triangle on that side and equal to that angle of depression ${20.25}^{\circ }$ ).

Here is an updated sketch with only the triangle information we need.

  $\begin{array}{l}\frac{21,120}{\sin {97.25}^{\circ }}=\frac{s}{\sin {62.5}^{\circ }}=\frac{f}{\sin {20.25}^{\circ }}\\ f=\frac{21,120\times \sin {20.5}^{\circ }}{\sin {97.25}^{\circ }}=7,369\text{ to the football field}\end{array}$

Thus,7,369 feet to the football field