Another problem about hyperbolas... T_T?

Question

Locating Whales

Hyperbolas can be used to locate objects underwater. To locate a whale in the ocean, two microphones are placed 8000 feet apart. One microphone picks up a whale noise 0.4 seconds after the second microphone picks up the same noise. The speed of sound in water is about 5000 feet per second.

a. How much farther from the whale is the first microphone?

b. Find an equation for the possible locations of the whale.

c. What is the closest distance that the whale could be to the second microphone?

d. Will the whale always be closer to the microphone that receives the signal first? Can the whale be on either branch of the hyperbola? Explain your reasoning.

Answer 1

a) 5000*,4 = 2000 feet, so a = 1000 feet

b) x^2/a2 -y2/b^2 =1, where;
where the two foci are located at (-c,0) and (c, 0) so c= 4000 ft
2a = difference in distances between foci and whale
b^2 = c^2-a^2

c) Closest when y = 0 which give x= abs(a)

d) y=+ or - b/a[sqrt ( x^2-a^2)], so if x < a, then y is imaginary. Therefore the region between x=a and x=-a must be excluded, that is, the whale can never be in that region. The whale will always be on the hyperbola branch that is associated with the mike that first receives the signal. The whale can be on either branch of the hyperbola, but never (obviously) on both at the same time. The distance from one foci to the whale will always be < or > than the distance from the other foci to the whale. The distances can never be equal. If they were, this technology would not work.

I believe I included all the info you need to answer the question completely. It would be nice if a diagram could be included.

Answer 2

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Answer 3

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Answer 4

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