Problem 49 - Precalculus Mathematics for Calculus

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 \(1000 \mathrm{mi}\). Will the beam strike the moon?

Step-by-Step Solution

Answer

(a) The beam diverges approximately 2094.4 miles. (b) No, the beam won't strike the moon.

Step 1: Understanding the Angle of Deviation

The laser beam deviates by an angle of \(0.5^{\circ}\) when directed toward the moon. This small angle can be considered as part of a right triangle where the hypotenuse is the direct path from Earth to the moon (240,000 mi) and the opposite side is the distance the beam strays from the target.

Step 2: Calculate the Divergence of the Beam

To find the divergence in miles, we use the formula for an arc length in degrees: \ \[\text{Arc Length} = \theta \times \frac{\pi}{180} \times r\] where \( \theta = 0.5^{\circ} \) and \( r = 240,000 \text{ mi} \). \ Substituting these values: \ \[\text{Arc Length} = 0.5 \times \frac{\pi}{180} \times 240,000 \approx 2094.4 \text{ mi} \] \ So, the beam strays approximately 2094.4 miles from its target.

Step 3: Compare Beam Divergence with the Moon's Radius

The moon has a radius of about 1000 miles. The beam's divergence of approximately 2094.4 miles is much larger than the moon's radius. This means the beam will not strike the moon as it diverges beyond the moon's edge.

Key Concepts

Angle of Deviation

The angle of deviation is a key concept in trigonometry, particularly when dealing with paths or trajectories that stray from the intended course. When a laser beam is aimed at the moon but deviates by an angle, such as the given \(0.5^{\circ}\), it means the beam's path has shifted slightly from its original direction. Understanding this angle involves visualizing a right triangle where:

• The angle of deviation is at one of the triangle's corners.
• The hypotenuse is the original direct path (Earth to the Moon, which is 240,000 miles in this case).
• The opposite side represents the distance the beam strays from the target.

This type of angular deviation can drastically affect the final position of the beam, as small angles can lead to significant shifts over large distances, such as between celestial bodies.

Arc Length Calculation

Arc length calculation is essential to determine how far a beam, or any line, has strayed from a path over a circular arc. In this scenario, when the laser deviates by an angle of \(0.5^{\circ}\), the formula for calculating the arc length in degrees is:\[\text{Arc Length} = \theta \times \frac{\pi}{180} \times r\]where:

• \(\theta = 0.5^{\circ}\) represents the angle in degrees.
• \(r = 240,000\text{ mi}\) is the radius or the direct distance to the moon.

When plugging these values into the formula, the result is approximately 2094.4 miles. This number represents how far the laser beam has deviated from its intended path towards the moon, thus helping us understand the extent of the angular deviation over a long distance.

Right Triangle

A right triangle is a fundamental structure used to solve problems involving angles and distances. In this exercise, the deviation of the laser beam can be modeled using a right triangle, which aids in understanding its stray from the path.In the triangle:

• The hypotenuse is the direct path from Earth to the moon (240,000 miles).
• The angle of \(0.5^{\circ}\) is the vertex angle adjacent to the hypotenuse.
• The opposite side (arc length of approximately 2094.4 miles) represents the linear distance the beam has deviated away from the target.

Right triangles allow us to apply trigonometric principles such as sine, cosine, and tangent, even though in this particular problem, the primary focus is on the relationship between the angle and the arc length derived using the circle's radius. Thus, the right triangle here serves as a conceptual tool to further understand angular deviation and path calculation.