Question

A hot air balloon appears as a blip on the air traffic controller’s radar screen. Hot air balloons are prohibited from flying within 25 miles radius of the airport. The polar grid shows the air traffic controller tracking the hot air balloon. (Radius = 10 miles) The grid shows the location of the hot air balloon at time t =0 minutes (when it first appeared on the radar screen) and time t=90 minutes later.


Express the two locations of the hot air balloon in polar coordinates. Is the hot air balloon in violation of the airport air space at either of these locations? Explain. (2 points)
Determine the rectangular coordinates of the hot air balloon at time t = 90 minutes. State the meaning of this answer in the context of the problem. (3 points)
The hot air balloon crashes 6 miles east and south of the air traffic controller’s tower. Determine the distance the balloon is from the control tower and if it has violated the airport air space. Justify your answer. (3 points)

Answer 1

a) The hot air balloon did not violate that air space in both locations.

b) The hot air balloon is 17.678 miles to the west and 17.678 miles to the north from the airport.

c) The crashed hot air balloon violated the airport air space. ($r \approx 8.485\,mi$, $r < 25\,mi$)

a) Polar coordinates of the hot air balloon are described solely by the distance from origin (airport) ($r$), in miles, and angle in standard position ($\theta$), in sexagesimal degrees, that is to say:

$z(t) = (r, \theta)$ (1)

Where $t$ is the time, in minutes.

The location of the hot air balloon at both instants is described below:

$z(0) = \left(37.5\,mi, \frac{5\pi}{6}\,rad \right)$, $z(90) = \left(25\,mi, \frac{\pi}{4}\,rad \right)$

In this case, the hot balloon violates the airport air space if and only if $r < 25\,mi$ and the hot air balloon does not violate that air space in both locations.

b) The equivalent polar coordinates of the hot air balloon in terms of polar components is described by the following formula:

$(x,y) = (r\cdot \cos \theta, r\cdot \sin \theta)$ (2)

Where:

• $x$ - Horizontal distance with respect to origin, in miles.
• $y$ - Vertical distance with respect to origin, in miles.

If we know that $r = 25\,mi$ and $\theta = \frac{\pi}{4}\,rad$, then the coordinates in rectangular form is:

$(x,y) = \left(25\cdot \cos \frac{\pi}{4}, 25\cdot \sin \frac{\pi}{4} \right)\,[mi]$

$(x,y) = (17.678, 17.678)\,[mi]$

The hot air balloon is 17.678 miles to the west and 17.678 miles to the north from the airport.

c) The distance of the crashed hot air balloon with respect to origin is determined by Pythagorean theorem: ($x = 6\,mi$, $y = -6\,mi$)

$r = \sqrt{(6\,mi)^{2}+(-6\,mi)^{2}}$

$r \approx 8.485\,mi$

Since $r < 25\,mi$, the crashed hot air balloon violated the airport air space.

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