Problem 29 - Algebra 2

Question

Automobile Design Suppose a windshield wiper arm has a length of 22 in and rotates through an angle of \(110^{\circ} .\) What distance does the tip of the wiper travel as it moves once across the windshield?

Step-by-Step Solution

Answer

The distance that the tip of the wiper travels across the windshield is equal to the arc length, which can be found by first converting the angle to radians and then multiplying the radius by this angle.

Step 1: Convert Degrees to Radians

Before calculations, convert the given angle from degrees to radians. This is because the formula for arc length requires the angle to be in radians. The conversion formula is \(Radians = Degrees \times \frac{\pi}{180}\). Use this formula to convert \(110^{\circ}\) to radians.

Step 2: Calculate Arc Length

The formula for arc length is \(Arc Length = radius \times angle\). After converting the angle to radians in step 1, we use the given length of the windshield wiper as the radius. Plug these numbers into the formula to find the arc length.

Step 3: Interpret the Result

The result obtained from step 2 is the distance that the tip of the wiper travels across the windshield in inches. It is the solution to the problem.

Key Concepts

Angle Conversion

When dealing with problems in trigonometry or related math fields, angle conversion is often necessary, particularly when you're calculating arc lengths or dealing with circular motion. It's common to encounter angles given in degrees, but for calculations involving circular measurements, it's more efficient to use radians. The conversion formula from degrees to radians is essential:

• To convert an angle from degrees to radians, multiply the degree measure by \(\frac{\pi}{180}\).
• For example, an angle of \(110^{\circ}\) becomes \(110 \times \frac{\pi}{180}\), which simplifies this to approximately \(1.92\) radians.

Aligning your measurements is crucial for accurate calculations, ensuring they're in the form required by standard geometric formulas.

Radian Measure

Radian measure is a crucial concept, especially in contexts involving circles or angles related to circular paths. Unlike degrees, which divide a circle into 360 equal parts, radians use the arc length of a circle with a radius of one, giving a more natural and mathematical representation of an angle.

• One full circle equals \(2\pi\) radians, simplifying the process for calculating portions of a circle.
• This measure is more compact and integrates seamlessly with many mathematical formulas, including those related to arc length and angular velocity.

Let's reconsider the windshield wiper problem: after converting \(110^{\circ}\) to radians as \(1.92\) radians, we can comfortably use this in further calculations for arc length.

Circular Motion

Circular motion is all about objects moving along a path defined by a circular trajectory. Understanding it involves dissecting the relationship between radius, angle, and arc length. It translates linear movements into curvilinear paths, commonly measured in radians.

• Key relationship: when an object rotates, the arc length it travels is calculated using \(\text{Arc Length} = \text{Radius} \times \text{Angle in Radians}\).
• In our windshield wiper scenario, with a 22-inch radius and an angle of \(1.92\) radians, the tip travels an arc length of approximately \(22 \times 1.92 = 42.24\) inches.

This principle highlights the interconnectedness between angle and distance—a foundational element in physics and engineering.

Windshield Wiper Geometry

Windshield wiper geometry may sound mundane, but it's an exciting example of practical applications of geometry in daily life. Every wipe of the wiper involves multiple calculations of arc lengths and angles.

• The wiper's path, forming a sector of a circle, translates linear rotational movements into swipes across the windshield.
• Understanding this involves calculating the arc length based on the wiper's length (radius) and the angle it covers on each swipe.

By employing the right formulas and careful conversion of measurement units, one can model and predict how much of the windshield each wipe will cover—a small yet effective illustration of geometry at work in automotive design.