Cosine function problem?

Question

A cyclist rides one lap of a circular track at a constant speed so that her distance, "d", from her starting point at any time, "t", after starting is:

d = 50 - 50 cos ( pi t / 30 )

(fifty minus fifty cosine of pi t divided by 30)

find:

a)time taken to complete one lap

b)radius of track

c)the maximum distance from the start

d)the length of the track

e) the distance from her starting position after:

-15 seconds

-40 seconds

Please explain answers and show working.

Answer 1

First, let's look at a cosine. It starts at +1 and goes to -1, then back to +1.

a. The period is how long it takes for the argument of the cos to go 2π. Thus,

2π = π·t / 30 so t = 60.

b. Radius is half of diameter, or furthest distance. This is then cos() = -1, or d = 50 - (-50), so d=100 and r = 50.

c. same as b. So, when you did b, they probably wanted you to say that 50 is the amplitude (number in front of the cos), thus the radius.

d. πd or 2πr is length (perimeter) of circle.

e. plug it in!