An airplane is flying at a speed of 340 mi/h at an altitude of one mile and passes directly over a radar stati

Question

An airplane is flying at a speed of 340 mi/h at an altitude of one mile and passes directly over a radar station at time t = 0.

There are 3 parts to this question. I found the first part but need the second two.

(a) Express the horizontal distance d (in miles) that the plane has flown as a function of t.
d(t) = 340t

(b) Express the distance s between the plane and the radar station as a function of d.
s(d) = ???

(c) Use composition to express s as a function of t.
s(t) = ???

Answer 1

Nice question!!
here's the solution:

(b) given that the altitude is 1 mile, at any point of time, the distance between the plane and the radar station is the hypotenuse of a right triangle where the other two sides are 1 mile and d miles respectively. hence,
s(d) = sqrt(d^2 + 1) ....[sqrt: square root]

(c) from the other equations 1 and 2,
s(t) = sqrt(340x340 t^2 +1)

In the above equation, as the time increases and becomes large, the equation can be approximated to
s(t) = 340 t

Answer 2

From the Theorem of Pythagoras:
s(d) = (1+d^2)^(1/2)

s(d) = (1+115,600*t^2)^0.5, s in miles, t in hours

The 1 is pretty much insignificant if t > 0.1 hr (6 min.) (error is .04% if you drop the 1)

Answer 3

(b) s^2 = d^2 + 1^2

s = sqrt(d^2 + 1)
s(d) = sqrt(d^2 + 1)

(c) substitute d with t where d(t) = 340t
s(t) = sqrt((340t)^2+1)