A cyclist rides one lap of a circular track at a constant speed so that the riders distance, d meters, from the starting point at any time, t seconds, after starting is: d= (50) - ( 50cos x (pi)t/30, find:

(a) the time taken to complete one lap
(b) the radius of the track
(c) the maximum distance from the start
(d) the length of the track
(e) the distance of the cyclist from the starting point at (i) 15 seconds (ii) 40 seconds
(f) the time (nearest second) at which the cyclist is 93n3 m from the starting point

To find the answers to the given questions, we need to use the equation for the rider's distance from the starting point at any time, which is given as:

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

Let's go through each question one by one:

(a) The time taken to complete one lap can be found by looking at the period of the cosine function in the equation. The period of a cosine function is given by the formula: period = (2 * pi) / coefficient of t. In this case, the coefficient of t is (pi / 30), so the period of the cosine function is:
period = (2 * pi) / (pi / 30)
= 60 seconds

(b) The radius of the track can be determined by considering that when the rider completes one full lap, the distance covered is equal to the circumference of the track (2 * pi * radius). So, the radius can be found by equating the distance of one lap (circumference) to 2 * pi * radius and solving for radius:
50 = 2 * pi * radius
radius = 50 / (2 * pi)
radius ≈ 7.9577 meters

(c) The maximum distance from the start occurs when the cosine function reaches its maximum value, which is 1. So, plugging in cos(0) = 1 into the equation, we have:
d = 50 - 50cos(0)
= 50 - 50(1)
= 0 meters

Therefore, the maximum distance from the starting point is 0 meters.

(d) The length of the track is equal to the distance covered by the rider when completing one full lap, which is the circumference of the track. So, the length of the track is given by:
length = 2 * pi * radius
= 2 * pi * 7.9577
≈ 50.1324 meters

(e) To find the distance of the cyclist from the starting point at a specific time, substitute the value of t into the equation:
(i) At 15 seconds:
d(15) = 50 - 50cos((pi * 15) / 30)

(ii) At 40 seconds:
d(40) = 50 - 50cos((pi * 40) / 30)

(f) To find the time at which the cyclist is 93.3 meters from the starting point, set the equation equal to 93.3 and solve for t:
93.3 = 50 - 50cos((pi * t) / 30)
cos((pi * t) / 30) = (50 - 93.3) / 50
cos((pi * t) / 30) = -0.866
Now, use the inverse cosine function to find t:
(pi * t) / 30 = arccos(-0.866)
t = (30 / pi) * arccos(-0.866)

Solve the equation to find the exact value for t, which can then be rounded to the nearest second.