Havok™, an Intel® company, was founded in Dublin, Ireland in1998, and it is the premier provider of interactive software andservices for digital media creators in the games and movieindustries. Havok works in partnership with the world’s best-knowngame developers, including Microsoft®, Nintendo®, and Sony®. Formore information about R- and TM -symbols please refer tothe Larry Donahue article. It is nothing to do with ourcourse. However, on a daily basis, Havok uses several techniquesthat appear as part of the College Algebra curriculum.

In Stealth Fighter 6 theguards are able to detect you, even if they cannot see you, byhearing the sound of the footsteps you make. Of course, the furtheraway you are from the guards, the less sound energy reaches them,and the quieter your footsteps sound. The way sound energydecreases over distance from the initial sound source obeysrational functions or inverse square law.

The energy generated by a sound is radiated outwards sphericallyfrom the source of the sound. Since the surface area of a sphere(4?r²) is proportional to the square of the radius, asthe emitted energy moves out from the source, it must spread outover an area that is proportional to the square of the radius(distance from the source). Hence, the energy passing through anyunit area is inversely proportional to the square of the distancefrom the point source (radius). In terms of sound energy (or signalstrength) this is represented using the following formula:

S = F/D² where S =sound energy heard, F = total energygenerated by the sound and D = distance(in meters) you are from the source of the sound.

Example

In Stealth Fighter 6 imagineyou had to sneak across a room behind the guards back. A guard canhear you if anything more than 1/10 (one tenth) of the footstepsound energy reaches them. What is the closest safe distance youcan approach them without being heard?

Solution:

S = F/ D^2

1/10 = 1/ D^2

D^2 = 10

D = sqrt(10)

D = 3.16 m

Therefore, if you move within 3.16 meters of the guardthey will hear you.

Playing Havok

You wish to cross the 11 m x 10 m courtyard without beingdetected by the two guards and escape through a door located on thewestern wall. Guard 1 is located in a tower at the northeast cornerof the yard and Guard 2 is located in the yard, in the southwestcorner, as shown in the figure below.

Guard 1 can hear you if anything more than 1/36 of the footstepsound energy reaches them; similarly Guard 2 can hear you ifanything more than 1/16 of the footstep sound energy reaches them(Guard 1 can hear more due to his elevated height in the tower).Assume for now that both guards are fixed at these locations andcannot move unless they detect you. Based on this information,answer the following questions.

1. Work out the maximum hearing range of both guards? (Hint: seeexample above)

For Guard 1 the maximum hearing range is __________________.

For Guard 2 the maximum hearing range is __________________.

2. Determine, if it exists, the safe distance between the twohearing zones of the guards for you to pass through undetected?(Hint: use Pythagorean Theorem or distance formula)

Distance between two red dots is:________________ .

We know that the radius of the hearing zones are: __ m for Guard1 and __ m for Guard 2.

Therefore the safe distance is:__________ .

3. Now assume that Guard 2 is on patrol. He walks a straightpatrol route up and down along the dotted line always remaining 3 mfrom the western wall. Determine the new maximum safe distancebetween the two hearing zones of the guards for you to passundetected? (Hint: use Pythagorean Theorem or distance formula)

Where is the Guard 2 located for the maximum distance betweenthe guards to occur? (at the northern or southernwall?)____________. In this case, the red dot for Guard 2 isshifted from location (3,3) to ____.

When this happens the distance between the guards is:________________.

Therefore, the maximum safe distance is:___________.

4. Check your classmates’ submissions and comment on twoposts.