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. 2. 3.
c. Calculate the distance of Cape Canaveral from the equator by calculator y when
d. I checked on the Internet, and Cape Canaveral is really 1969 miles north of the equator. That’s 3,169 km. Did your
model give a reasonable accurate answer? How much is it off by?
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 seconds the current is at its maximum, amperes.
a. Write an equation expressing current in terms of time
b. What is the current when ?
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 meters? At meters?
5. An old rock formation is warped into the shape of a sinusoid. Over the centuries, the top has eroded away, leaving the ground with a flat surface from which various layers of rock are cropping out.
Since you have studied sinusoids, the geologists call upon you to predict the depth of a particular formation at various points. You construct an x-axis along the ground and a y-axis at the edge of an outcropping, as shown. A hole drilled at meters shows that the top of the formation is 90 meters deep at that point.
a. Write an equation expressing the y-coordinate of the formation in terms of x.
b. If a hole were drilled to the top of the formation at , how deep would it be?
c. What is the maximum depth of the top of the formation, and what is the value of x where it reaches this depth?
d. How high above the present ground level did the formation go before it eroded away?
Hints on number 5:
Since you can’t solve for a ord, calculate the period and phase shift. Now fill in the equation for the sinusoidal function, letting a represent the amplitude and d represent the vertical shift. We’ll just use those letters since we don’t know their values.
We have one equation with 4 variables. But we know 2 other points, (0, 0) and (100, -90). We can substitute the first point in for x and y to create an equation with only two unknowns, then substitute (100, -90) for x and y to create a second equation with two unknowns. Now we can solve the system of two equations and two unknowns to find a and d, with a little help from our friend the unit circle.