Precalculus Honors Worksheet
Sinusoidal Applications
Name:
1. Electrical Current and Voltage Problem
The electricity supplied to your house is called alternating current (AC) because the
current varies sinusoidally with time. The voltage that causes the current to flow also
varies sinusoidally with time. Both current and voltage have a frequency of 60
cycles per second in the United States, but have different phase displacements.
a) Suppose that the current is at its maximum of 5 amperes (A) when t = 0 seconds.
Find a particular equation for this sinusoid.
b) As shown in the figure, the voltage leads the current by 0.003 seconds, meaning
that it reaches a maximum 0.003 seconds before the current does. (“Leading”
corresponds to a negative phase displacement and “lagging” corresponds to a
positive phase displacement.) If the peak voltage is 180 volts (V) find a particular
equation for voltage as a function of time. (Note that the 115 V supplied to your
house is an average value, whereas the 180 V is an instantaneous peak voltage.)
c) What is the voltage at the time the current is a maximum?
d) What is the current at the time the voltage is a maximum?
e) What is the first positive value of t at which v reaches 170?
Answers: 1a) i = 5 cos (120π t ) b) v = 180 cos (120π ( t + 0.003)) c) v(0) = 76.642 v d) i(0.003)=2.1288 a e) t=0.0127seconds
2. Rock Formation Problem
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 wich various rock
formations are cropping out. Because you have studied sinuosids, the geologists call on
you to predict the depth of a particular stratum (boundary surface) 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 x = 100 meters shows that the top of the stratum is 90 meters deep at that
point.
a) Write a particular equation expressing y as a function of x.
b) If a hole were drilled to the stratum at x = 510 meters, how deep would it be?
c) What is the maximum depth of the stratum, and what is the smallest positive
value of x at which it reaches the maximum depth.
d) How high above the present ground level did the stratum go before it eroded away?
e) For what values of x between 0 and 800 meters is the stratum within 120 meter of
the surface?
f) The geologists decide to drill holes to the stratum every 50 meters from x = 50
meters to x = 750 meters searching for valuable minerals. Each hole costs $75 per
meter to drill. What is the total cost of drilling the holes
π
( x + 200 ) b) -240.9607 m c) (400,270)
600
e) 0 ≤ x ≤ 131.9802 and 668.0197 ≤ x ≤ 800 f) $196,804.56
Answers: 2a) y = −90 + 180 cos
d) 90 m