Download Sinusoidal Application Problems – Part 2

Sinusoidal Application Problems – Part 2
When a spaceship is fired into orbit from a site such as Cape Canaveral, which is not on the equator, it goes into an
orbit that takes it alternately north and south of the equator. Its distance from the equator is approximately sinusoidal
function of time.
Suppose that a spaceship is fired into orbit from Cape
Canaveral. Ten minutes after it leaves the Cape, it reaches its
farthest distance north of the equator, 4000 kilometers. Half a
cycle later it reaches its farthest distance south of the equator
(on the other side of the Earth, of course!) also 4000 kilometers.
The spaceship completes an orbit once every 90 minutes.
Let y be the number of kilometers the spaceship is north of the
equator (you may consider distances south of the equator to be
negative). Let t be the number of minutes that have elapsed
since liftoff.
a. Find the equation of the spaceship’s distance from the equator over time and sketch its graph.
b. Use the equation to predict the distance of the spaceship form the equator when:
1. t = 25
3. t = 163
2. t = 41
c. Calculate the distance of Cape Canaveral from the equator by calculator y when t = 0
d. See if you can find how far Cape Canaveral really is from the equator to see if the model gives reasonable accurate
answers.
2. The electricity supplied to your house is called “alternating current” because the current varies sinusoidally with time.
The frequency of the sinusoid is 60 cycles per second. Suppose that at time t = 0 seconds the current is at its maximum,
i = 5 amperes.
a. Write an equation expressing current in terms of time
b. What is the current when t = 0.01 ?
3. The alternating electrical current supplied to your house is created by an alternating electrical potential, or “voltage,”
which also has a frequency of 60 cycles per second. For reasons you will learn when you study electricity (next year) the
voltage usually reaches a peak slightly before the current does. Since the voltage peak occurs before the current peak, the
voltage is said to “lead” the current, or the current “lags” the voltage. Leading corresponds to a negative phase
displacement and lagging corresponds to a positive phase shift.
a. Suppose that the peak voltage is 180 volts and that the voltage leads the current by 0.003 seconds. Write an equation
expressing voltage in terms of time.
b. Predict the voltage when the current is a maximum.
4. A portion of a roller coaster track is to be built in the shape of a sinusoid. You have been hired to calculate the lengths
of the vertical timber supports to be used.
a. The high and low points on the track are separated by 50 meters horizontally and by 30 meters vertically. The low point
is 3 meters below the ground. Letting y be the number of meters the track is above ground and x be the number of meters
horizontally from the high point, write an equation expressing y in terms of x.
b. How long is the vertical timber at the high point? At x = 4 meters? At x = 32 meters?