Superlesson
Project 6-3
Foundations of Algebra and Geometry
Chapter 6, Using Ratios to Compare
Trigonometry can be used to make indirect measurements, including measurements of both the speed of sound and the speed of light.
Part A, Trigonometric Ratios
1. An observer at A saw lightning strike the top of the spire on the Petronas Towers, the world's tallest building, located in Kuala Lumpur, Malaysia .
The sound of thunder from the strike reached the observer's ears 5 seconds later.
a. How tall are the Petronas Towers? Round to the nearest foot? (Hint: 1 meter = 3.28084 feet)
b. The observer measured the angle to the top of the building at 15° from the horizontal. Use trigonometry to find distance AC. Round to the nearest foot.
c. Use the Pythagorean Theorem to find AB, the distance the sound of thunder traveled to reach the observer's ears.
d. Find the speed of the sound of the thunder that the observer heard.
Part C, Making Connections
2. During the seventeenth century, the Danish astronomer Ole Roemer invented the first practical method for measuring the speed of light. The following problem is based on Roemer's method.
The figure shows Earth (E), the Sun (S), and Jupiter at
two positions in its orbit around the Sun (J1 and J2).
a. Find distance SJ1 and SJ2, Jupiter's mean distance from the Sun, in miles. (NOTE: 1 AU = 92,831,059.004 miles)
b. When Earth, Jupiter, and the Sun form a right triangle, as in the figure,
. Find EJ2, the distance from Earth to Jupiter at such a time.
c. One of Jupiter's moons was behind Jupiter and therefore not visible from Earth. If Jupiter had been at J1, the moon would have appeared to an astronomer on Earth to emerge from behind Jupiter at 2:49:30 A.M. But Jupiter was at J2, which is closer to Earth than J1. The light from the moon reached Earth faster than it would have from J1 because it didn't have to travel as far. In fact, it reached Earth at 2:42:00 A.M. How much sooner did it arrive than it would have from J1? Give your answer in seconds.
d. How much closer to Earth is Jupiter at J2 than it is at J 1?
e. At what rate of speed must light travel to cover the extra distance from Earth to J1 in the time from 2:42:00 A.M. to 2:49:30 A.M.?