The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. Express a rider’s height above ground as a function of time in minutes.
With a diameter of 135 m, the wheel has a radius of 67.5 m. The height will oscillate with amplitude 67.5 m above and below the center.
Passengers board 2 m above ground level, so the center of the wheel must be located
m above ground level. The midline of the oscillation will be at 69.5 m.
The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.
Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve.
Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of
The function
is odd, so its graph is symmetric about the origin. The function
is even, so its graph is symmetric about the
y -axis.
The graph of a sinusoidal function has the same general shape as a sine or cosine function.
In the general formula for a sinusoidal function, the period is
See
[link] .
In the general formula for a sinusoidal function,
represents amplitude. If
the function is stretched, whereas if
the function is compressed. See
[link] .
The value
in the general formula for a sinusoidal function indicates the phase shift. See
[link] .
The value
in the general formula for a sinusoidal function indicates the vertical shift from the midline. See
[link] .
Combinations of variations of sinusoidal functions can be detected from an equation. See
[link] .
The equation for a sinusoidal function can be determined from a graph. See
[link] and
[link] .
A function can be graphed by identifying its amplitude and period. See
[link] and
[link] .
A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. See
[link] .
Sinusoidal functions can be used to solve real-world problems. See
[link] ,
[link] , and
[link] .
Section exercises
Verbal
Why are the sine and cosine functions called periodic functions?
The sine and cosine functions have the property that
for a certain
This means that the function values repeat for every
units on the
x -axis.
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ...
Step 2: Find each score's deviation from the mean. ...
Step 3: Square each deviation from the mean. ...
Step 4: Find the sum of squares. ...
Step 5: Divide the sum of squares by n – 1 or N.
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400
a. what is the probability of getting more than 12,000 hits?
b. what is the probability of getting fewer than 9,000 hits?
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400.
a. What is the probability of getting more than 12,000 hits