Pre-Calc/Calc Question on Functions?

Question

Hello...can someone help me with this hw problem? "An airplane is flying at a speed of 300 mph at an altitude of one mile and passes directly over a radar station at time t=0....express the horizontal distance that the plane has flown as a function of t...express the distance s between the plane and radar station as a function of d. Thanks!

For the last part of the question, how would I come up with the formula to express the distance? I still am cloudy on that.

One more quick question...how would I use composition to express s as a function of t?

One more quick question...how would I use composition to express s as a function of t?

Answer 1

x = horizontal distance in miles
1 = height in miles
d = distance from ground to plane in miles
d ² = 1 ² + x ²

300 miles<---> 1 hour
x miles<--> x / 300 hours

d ² = 1 + x ² / (300) ²

Answer 2

the horizontal distance d the plane has flown can be given as

d(t) = 300t

the distance s between the plane and radar station can be given by

s(t) = ( d(t)^2 + 1)^1/2
s(t) = ( 90000t^2 + 1)^1/2

and remember t is in hours.

Additional Details

It is the pythagorean theorem. The altitude of the plane is one of the sides of the triangle and the horizontal distance the plane has flown is the other side and forms a right angle to the altitude. The distance from the plane to the radar station is the hypoteneuse of the right triangle.

Answer 3

horizontal distance is h(t) = 300 * t

distance from radar station is d = Sqrt(1 + (300 * t)²)
===Sqrt( (vertical distance)² + (horizontal distance)²)

Not sure what else you really need other than that.

Answer 4

Okay. You have to get used to asking yourself, what different functions do I have here. In this case you obviously (I hope) have a csc and a sqrt. Let f(x) = csc x and g(x) = sqrt(x). Now we want the csc in the sqrt, right? so we want sqrt(f(x)) and we have u(t) = sqrt(f(t)) and since g(x) is the sqrt u(t) =g(f(t)). or u(t) = g circle f of t.