The height of a rider on a Ferris wheel is given by h(t)=12-10c0s(2pi)(t) meters, where t gives time, in minutes of the ride.
a. Find the amplitude, midline, and period of the function h.
b. During the first two minutes of the ride, find the times when the rider has a height of 10 meters.
To find the amplitude, midline, and period of the function h(t)=12-10cos(2πt), let's break down each component:
a. Amplitude: The amplitude of a trigonometric function is the absolute value of the coefficient of the trigonometric term. In this case, the coefficient of the cosine term is 10. So, the amplitude is |10| = 10.
b. Midline: The midline of a trigonometric function is the vertical shift of the graph. In this case, the equation is h(t)=12-10cos(2πt), so the midline is the constant term, which is 12.
c. Period: The period of a trigonometric function represents the length of one complete cycle. The period can be found by dividing 2π by the coefficient of the variable, t. In this case, the coefficient of t is 2π, so the period is 2π/(2π) = 1.
Therefore, the amplitude is 10, the midline is 12, and the period is 1.
To find the times when the rider has a height of 10 meters during the first two minutes of the ride, we substitute h(t) = 10 and solve for t:
h(t) = 10
12 - 10cos(2πt) = 10
Subtracting 12 from both sides:
-10cos(2πt) = -2
Dividing by -10:
cos(2πt) = 0.2
Now, we need to find the values of t that satisfy this equation within the given time interval of the first two minutes (0 ≤ t ≤ 2).
To find the values of t, we can use the inverse cosine function (also known as arccosine or cos^-1) to isolate t:
2πt = cos^(-1)(0.2)
Now, we solve for t:
t = cos^(-1)(0.2) / (2π)
Using a calculator to evaluate the inverse cosine of 0.2, we find:
t ≈ 0.997, 1.262
Therefore, during the first two minutes of the ride, the times when the rider has a height of 10 meters are approximately 0.997 minutes and 1.262 minutes.