Chapter 1.4, Problem 69E

(a)

To determine

The diagram that gives the visual representation.

Answer to Problem 69E

The transmitter and the receiver are separated by the distance d.

Explanation of Solution

Given:

The balloon carrying the transmitter moving vertically upwards from a point 3000 feet from the receiving station.

Calculation:

Consider a balloon carrying a transmitter moving vertically upwards from a point 3000 feet from the receiving station.

Let h be the height of balloon and d is the distance between the balloon and receiving station.

(a) Drawing the diagram for the situation.

Conclusion:

The transmitter and the receiver are separated by the distance d.

(b)

To determine

The height of the balloon as the function of d.

Answer to Problem 69E

The height of the balloon is $h=\sqrt{d^2-{3000}^2}$

Explanation of Solution

Given:

The balloon carrying the transmitter moving vertically upwards from a point 3000 feet from the receiving station.

Formula used:

The Pythagoras theorem is used.

Calculation:

Calculating the function of height $h$ in terms of $d$

Using Pythagoras theorem,

  $\begin{array}{l}{d}^2=h^2+{3000}^2\\ h^2=d^2-{3000}^2\\ h=\sqrt{d^2-{3000}^2}\end{array}$

This is the required function

To determine the domain of the function, the radical should be non-negative. To calculate as below:

  $\sqrt{d^2-{3000}^2}\ge 0$

Squaring both sides

  $\begin{array}{l}{d}^2-{3000}^2>0\\ d^2>{3000}^2\\ d\ge 3000\end{array}$

Hence, the domain of this function is all real values of d where $d\ge 3000$

Conclusion:

The height of the balloon is $h=\sqrt{d^2-{3000}^2}$ and the domain of this function is all real values of d where $d\ge 3000$