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Chapter 1: Problem 36

The average distance to the Moon is \(384,000 \mathrm{~km}\), and the Moon subtends an angle of \(1 / 2^{\circ}\). Use this information to calculate the diameter of the Moon in kilometers.

Short Answer

Expert verified
The diameter of the Moon is approximately 602.3 kilometers

Step by step solution

01

Convert degrees to radians

The subtended angle needs to be in radians for the small angle approximation, which states that for small angles, the angle in radians is approximately equal to its sine. To convert degrees to radians, use the conversion factor \( \frac {π}{180} \) radians per degree. Thus, \( θ = 1 / 2 ^{\circ} * \frac {π}{180} = 1.57 \times 10^{-3} \) radians.
02

Apply small angle approximation

The small-angle approximation states that the angle \( θ \), when expressed in radians, approximates the sine of the angle for small values. Therefore, \( \sin(θ) ≈ θ \) . In this scenario, the distance to the moon is in the line of sight (the hypotenuse), and the diameter of the moon represents the opposite side in a right triangle setup. Therefore, \( \sin(θ) = \frac {diameter}{distance} \). Applying the small angle approximation, \( θ = \frac {diameter}{distance} \)
03

Calculate the diameter

Rearrange the formula to solve for the diameter of the moon. This gives \( diameter = θ * distance \). Substitute the given values into the equation: \( diameter = 1.57 \times 10^{-3} radians * 384,000 km = 602.3 km \). The diameter is rounded to one decimal place for accuracy

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Most popular questions from this chapter

Give the word or phrase that corresponds to the following standard abbreviations: (a) \(\mathrm{km}\), (b) \(\mathrm{cm}\), (c) s, (d) \(\mathrm{km} / \mathrm{s}\), (e) \(\mathrm{mi} / \mathrm{h}\), (f) \(\mathrm{m}\), (g) \(\mathrm{m} / \mathrm{s}\), (h) h, (i) \(\mathrm{y}\), (j) g, (k) \(\mathrm{kg}\). Which of these are units of speed? (Hint: You may have to refer to a dictionary. All of these abbreviations should be part of your working vocabulary.)

The age of the universe is about \(13.7\) billion years. What is this age in seconds? Use powers-of-ten notation.

What are degrees, arcminutes, and arcseconds used for? What are the relationships among these units of measure?

A reporter once described a light-year as "the time it takes light to reach us traveling at the speed of light." How would you correct this statement?

What is the difference between a theory and a law of physics?
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