Problem 47 - Algebra and Trigonometry

Question

A laser beam is to be directed toward the center of the moon, but the beam strays \(0.5^{\circ}\) from its intended path. (a) How far has the beam diverged from its assigned target when it reaches the moon? (The distance from the earth to the moon is 240,000 mi.) (b) The radius of the moon is about 1000 mi. Will the beam strike the moon?

Step-by-Step Solution

Answer

The beam diverges 2094.38 miles and does not strike the moon.

Step 1: Understand the Given Data

You are given that the laser beam is to be directed at the moon but strays by \(0.5^{\circ}\) from its intended path. The distance from Earth to the moon is 240,000 miles. We need to find how far the beam has diverged and if it will still strike the moon, knowing that the moon's radius is 1000 miles.

Step 2: Calculate Divergence Distance

To find out how far the beam deviates in miles from its intended path, use the formula for arc length, which is given by:\[L = r \times \theta\]where \(L\) is the arc length (the divergence), \(r = 240,000\) miles is the radius or distance to the moon, and \(\theta\) is the angle in radians. First, convert \(0.5^{\circ}\) to radians: \[\theta = 0.5 \times \frac{\pi}{180} \approx 0.0087266 \text{ radians}\]Now find the divergence:\[L = 240,000 \times 0.0087266 \approx 2094.38 \text{ miles}\]

Step 3: Determine if the Beam Strikes the Moon

The beam has diverged approximately 2094.38 miles from the intended target. Since the radius of the moon is 1000 miles, the beam deviated more than half the full moon's width (2000 miles). Hence, it will miss the moon completely.

Key Concepts

Angle Conversion

In trigonometry and geometry, angles are often presented in degrees, but calculations, especially those involving arcs and circles, are more easily performed in radians. Converting between these two units is a foundational skill. One degree is defined as 1/360 of a full rotation, while a radian is based on the radius of a circle.

• Conversion Formula: To convert degrees to radians, use the formula \( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \).
• Practical Example: In the given problem, the beam deviates by \( 0.5^{\circ} \), which can be converted to radians as \( 0.5 \times \frac{\pi}{180} \approx 0.0087266 \) radians.

It's important to become comfortable with this conversion because many trigonometric functions and physics calculations default to radians. For this specific exercise, converting an angle to radians is necessary to subsequently calculate the arc length.

Arc Length

The arc length of a circle is a measure of the distance between two points along the circle’s circumference. Understanding how to calculate this is crucial, particularly in applications involving circular motion, such as this exercise. The main formula for arc length \( L \) is:

• \( L = r \times \theta \)

where \( r \) is the radius, and \( \theta \) is the angle subtended by the arc in radians.

Why Arc Length Matters

The arc length tells us exactly how far a point on a circle has traveled. In this problem, the laser's path deviation from its intended target on the moon is calculated as an arc length. Given the radius of 240,000 miles (distance to the moon) and an angle of 0.0087266 radians, the arc length divergence is:

• \( L = 240,000 \times 0.0087266 \approx 2094.38 \) miles

This calculation reveals how far off the beam is from its intended mark due to the angular deviation.

Radians

Radians provide a natural measure of angle for mathematics and physical sciences. Unlike degrees, which divide a circle into 360 parts, radians relate directly to the dimensions of the circle through the arc.

Understanding Radians

One radian is the angle created by taking the radius and wrapping it along the circle's edge. A full circle encompasses \( 2\pi \) radians, equivalent to 360 degrees.

• Usage: Radians are used extensively because they simplify many mathematical concepts and equations. For instance, the circumference of a circle \( (C = 2\pi r) \), connections between angles and trigonometric functions, and calculus operations are simpler with radians.

In this exercise, the small deviation angle of the laser, \( 0.5^{\circ} \) converted into radians, aids the calculation of the actual physical divergence.

Distance Calculation

Determining the actual distance an object has traveled, especially in space, involves various factors including angles and dimensions of paths. In exercises like this one, focusing on accurate distance assessment is key.

Applying Distance Calculations

Distance calculations in the trigonometric application can be illustrated through scenarios like a beam missing its target. Here, the laser beam initially aimed at the moon deviates because of a small angular error.

• Using Arc Length: Distance off-target can be measured by arc length, with the formula \( L = r \times \theta \).
• Example Outcome: The laser beam was calculated to have diverged approximately 2094.38 miles from its intended path.
• Test Against Target Size: Knowing the moon's radius is 1000 miles helps determine the beam's deviation relative to the potential target (the moon). Since 2094.38 miles exceed the diameter of the moon (2000 miles), the beam indeed misses the moon.

This systematic approach shows how angle errors translate into real-world distance discrepancies, making these calculations fundamental to precision targeting in navigation and astronomy.