Problem 109 - A Graphical Approach to Precalculus with Limits

Question

The formula \(\omega=\frac{\theta}{t}\) can be rewritten as \(\theta=\) wt. Substituting wt for \(\theta\) changes \(s=r \theta\) to \(s=r \omega t\). Use the formula \(s=r \omega t\) to find the value of the missing variable. \(r=6\) centimeters, \(\omega=\frac{\pi}{3}\) radians per second, \(t=9\) seconds

Step-by-Step Solution

Answer

The arc length, \(s\), is \(18\pi\) centimeters.

Step 1: Understand the Problem

We are given the formula \(s = r \omega t\) and the parameters \(r = 6\) cm, \(\omega = \frac{\pi}{3}\) radians per second, and \(t = 9\) seconds. Our task is to find the value of \(s\), which represents the arc length.

Step 2: Substitute Values into the Formula

Insert the given values into the formula \(s = r \omega t\). This means substituting \(r = 6\), \(\omega = \frac{\pi}{3}\), and \(t = 9\) into the equation. The equation becomes: \[ s = 6 \times \left( \frac{\pi}{3} \right) \times 9 \]

Step 3: Simplify the Equation

Perform the multiplication step by step: \[ s = 6 \times \frac{\pi}{3} \times 9 = \frac{6 \times 9 \times \pi}{3} \]

Step 4: Complete the Multiplication

Calculate \(6 \times 9\) which equals 54. So our equation becomes \[ s = \frac{54 \pi}{3} \]

Step 5: Simplify the Result

Divide 54 by 3 to simplify the equation. This yields: \[ s = 18 \pi \] This is the arc length in centimeters.

Key Concepts

Arc Length

The arc length is the distance along the curved path of a circle's circumference, commonly represented by the symbol \(s\). If you think about tracing the edge of a circular pizza slice, what you are measuring is the arc length of that pizza. Arc length is crucial in understanding circular motion and various applications in engineering and physics.

The formula we use to find arc length is \(s = r \theta\), where \(r\) is the radius of the circle and \(\theta\) is the angle in radians that the arc subtends at the center of the circle. This shows that the arc length is a product of the radius and the angle covered in radians, making arc length a part of larger calculations involving circular objects or motions.

Radians

Radians are a way of measuring angles, used primarily in mathematics and physics. Unlike degrees, which divide a circle into 360 parts, radians turn the circle into 2π parts. Essentially, 1 radian is the angle created when the radius is wrapped along the circle's edge, covering a distance equal to the radius itself.

• 1 full circle = 2π radians
• 180 degrees = π radians
• 1 radian ≈ 57.3 degrees

Radians are particularly useful in trigonometry because they make many mathematical formulas much simpler and more natural. When angles are in radians, various trigonometric identities and properties are easier to apply.

Angular Velocity

Angular velocity, denoted by \(\omega\) (the Greek letter omega), measures how fast an object rotates around a circle. It is often described in terms of radians per second. Angular velocity gives us a sense of how quickly the angle (in radians) is changing as the object moves along the circular path.

The formula for angular velocity is \(\omega = \frac{\theta}{t}\), where \(\theta\) is the angular displacement, and \(t\) is the time taken for that displacement. Angular velocity is crucial in mechanical and aerospace applications, where understanding and predicting rotational movements are necessary.

• Higher angular velocity means faster rotation.
• Low angular velocity means slower rotation.

Formula Substitution

Formula substitution involves replacing variables in equations with their given values to compute unknown quantities. It is a standard mathematical technique used to solve equations efficiently. In the context of the exercise provided, it takes on a key role.

We start with the equation \(s = r \theta\) and substitute \(\theta\) with \(\omega t\), resulting in \(s = r \omega t\). This modification accounts for scenarios where the angle covered is not directly given, but can instead be derived from other known quantities like angular velocity and time. Through careful substitution:

• Identify the known variables and their values.
• Make a calculated substitution to rewrite the equation.
• Perform arithmetic to find the unknown variable, in this case, the arc length.

This method simplifies complex equations and makes them more practical for solving real-world problems.