The third-tallest Ferris Wheel in the world is the London Eye in England.The height (in metres) of a rider on the London Eye after t minutes can be described by the function h(t) = 67sin [12(t + 0.0223)] + 70.

a. At what time(s) will the rider be at the bottom of the Ferris wheel?
b. How long does it take for the Ferris wheel to go through one rotation?

(a) when sin(12(t + 0.223)) = -1

That is, when 12(t + 0.223) = 3π/2
(b) the period of sin(kt) is 2π/k

To determine the time(s) when the rider is at the bottom of the Ferris wheel, we need to find the values of t that make h(t) equal to the lowest point on the wheel.

a. The function h(t) represents the height (in meters) of the rider on the Ferris wheel at time t. In this case, the lowest point of the Ferris wheel occurs when the rider is at a height of 0 meters.

Setting h(t) to 0, we can solve for t:

0 = 67sin[12(t + 0.0223)] + 70

Subtracting 70 from both sides:

-70 = 67sin[12(t + 0.0223)]

Dividing both sides by 67:

-1.0448 = sin[12(t + 0.0223)]

To find the values of t that satisfy this equation, we can take the inverse sine (or arcsine) of both sides:

arcsin(-1.0448) = 12(t + 0.0223)

Using a scientific calculator, you can find that the arcsine of -1.0448 is approximately -90.69 degrees or -1.5807 radians.

Next, we divide by 12:

-0.1317 = t + 0.0223

Subtracting 0.0223 from both sides:

t = -0.1317 - 0.0223

Simplifying:

t = -0.154

So, one time when the rider is at the bottom of the Ferris wheel is at approximately t = -0.154 minutes.

Note that the negative time indicates that the rider was at the bottom of the Ferris wheel in the past, 0 represents the initial position, and positive values of t indicate future time.

To determine if there are any other times when the rider is at the bottom of the Ferris wheel, we can consider the periodic nature of the sine function. Since the lowest point (0 meters) occurs twice per cycle of the sine function, we can add one complete period to our initial time of t = -0.154 minutes to find the second time when the rider is at the bottom.

The period of a trigonometric function is determined by the coefficient inside the function. In this case, the coefficient is 12. The period (T) of the trigonometric function is given by T = 2π/|coefficient|.

T = 2π/|12| = π/6

Therefore, the time it takes for the Ferris wheel to complete one rotation is approximately π/6 minutes. Adding this time to our initial value, -0.154 minutes, we get:

t = -0.154 + π/6

Simplifying:

t ≈ 0.154 + 0.5236

t ≈ 0.6776

So, the second time when the rider is at the bottom of the Ferris wheel is approximately t = 0.678 minutes.

b. The time it takes for the Ferris wheel to go through one rotation is approximately π/6 minutes.