Ch. 3 - Radian Measure and The Unit Circle

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 3 - Radian Measure and The Unit Circle

Problem 23

Chapter 4, Problem 23

Distance between Cities Find the distance in kilometers between each pair of cities, assuming they lie on the same north-south line. Assume the radius of Earth is 6400 km. See Example 2. Panama City, Panama, 9° N, and Pittsburgh, Pennsylvania, 40° N

Verified step by step guidance

Identify the latitudes of the two cities: Panama City at 9° N and Pittsburgh at 40° N. Since they lie on the same north-south line, the difference in latitude will determine the distance between them.

Calculate the difference in latitude between the two cities: \(\Delta \theta = 40^\circ - 9^\circ\).

Convert the difference in latitude from degrees to radians because the arc length formula requires the angle in radians. Use the conversion formula: \(\text{radians} = \Delta \theta \times \frac{\pi}{180}\).

Use the arc length formula to find the distance along the Earth's surface: \(\text{distance} = R \times \text{radians}\), where \(R = 6400\) km is the radius of the Earth.

Substitute the values into the formula and simplify to express the distance between Panama City and Pittsburgh in kilometers.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Great Circle Distance on a Sphere

The shortest distance between two points on the surface of a sphere lies along the great circle connecting them. For points on the same meridian (north-south line), this distance corresponds to the arc length between their latitudes on the Earth's surface.

Arc Length Calculation Using Central Angle

The distance between two points on a circle can be found by multiplying the radius by the central angle (in radians) between them. Here, the central angle is the difference in latitude degrees converted to radians, and the radius is the Earth's radius.

Conversion Between Degrees and Radians

Angles measured in degrees must be converted to radians for use in arc length formulas. The conversion is done by multiplying degrees by π/180, since 180 degrees equals π radians.

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Textbook Question

Find each exact function value. See Example 2.

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Textbook Question

Distance between Cities Find the distance in kilometers between each pair of cities, assuming they lie on the same north-south line. Assume the radius of Earth is 6400 km. See Example 2.

Farmersville, California, 36° N, and Penticton, British Columbia, 49° N

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Textbook Question

Use the formula v = r ω to find the value of the missing variable.

v = 9 m per sec , r = 5 m

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Textbook Question

Use the formula v = r ω to find the value of the missing variable.

v = 12 m per sec, ω = 3π/2 radians per sec

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