Problem 108 - A Graphical Approach to Precalculus with Limits

Question

Use the formula \(\omega=\frac{\theta}{t}\) to find the value of the missing variable. Round to the nearest thousandth. \(\omega=0.90674\) radian per minute, \(t=11.876\) minutes

Step-by-Step Solution

Answer

The value of \( \theta \) is 10.771 radians.

Step 1: Understand the Given Formula

The formula given is \( \omega = \frac{\theta}{t} \), where \( \omega \) is the angular velocity in radians per minute, \( \theta \) is the angular displacement in radians and \( t \) is the time in minutes. We need to find \( \theta \).

Step 2: Rearrange the Formula to Solve for Theta

Rearrange the formula to solve for \( \theta \). Start by multiplying both sides by \( t \) to isolate \( \theta \) on one side of the equation: \( \theta = \omega \times t \).

Step 3: Substitute the Known Values

Substitute the given values into the rearranged formula: \( \theta = 0.90674 \times 11.876 \).

Step 4: Perform the Calculation

Calculate \( \theta \) by multiplying \( 0.90674 \) and \( 11.876 \). \(\theta = 10.77111904 \).

Step 5: Round the Result to the Nearest Thousandth

Round \( \theta \) to the nearest thousandth. The value \( 10.77111904 \) rounds to \( 10.771 \).

Key Concepts

Radian Measure

Radian measure is a way to quantify angles based on the radius of a circle. Instead of using degrees, which divide a circle into 360 parts, radian measure considers the arc length on the unit circle. Here's how it works: if you take the radius of a circle and wrap it along the circle's edge, the angle formed at the circle's center is approximately 57.3 degrees, which equals one radian. This measure is useful because it allows for more natural calculations in mathematics and physics.

Radian measure is directly related to the circle's geometry. In a complete circle, there are 2π radians, which is equivalent to 360 degrees. This relationship comes in handy when converting between degrees and radians:

• To convert from degrees to radians, use the formula: \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \)
• To convert from radians to degrees, utilize: \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \)

Understanding radian measure gives a versatile tool when dealing with circular motion, as it integrates naturally with other physical equations.

Angular Displacement

Angular displacement refers to the angle, usually measured in radians, through which a point or line has been rotated in a specified sense around a specified axis. It's an important concept in physics and engineering when dealing with the motion of bodies. Unlike linear displacement, which measures distance, angular displacement measures the change in the orientation of an object.

The calculation of angular displacement often involves the formula:
\( \theta = \omega \times t \)
where \( \theta \) is the angular displacement, \( \omega \) is the angular velocity, and \( t \) is the time over which the rotation occurs.

This relationship is derived from the definition of angular velocity, which is the rate of change of angular displacement over time. This straightforward formula allows us to calculate how much an object has rotated over a given period. For example, in the provided exercise, the angular displacement \( \theta \) was calculated by multiplying the angular velocity with time, yielding a result of 10.771 radians.

Time Calculation

Time calculation plays a pivotal role when determining angular displacement and investigating circular motion. When given two out of the three variables (angular velocity \( \omega \,\), angular displacement \( \theta \,\), and time \( t \)), one can easily find the missing value using the formula:
\( \omega = \frac{\theta}{t} \)

In the realm of rotational dynamics, time calculation often revolves around rearranging this formula depending on the missing variable.

• To find time (\( t \)), rearrange the formula to: \( t = \frac{\theta}{\omega} \).
• To find angular displacement (\( \theta \)), use: \( \theta = \omega \times t \).

This flexibility makes it easy to solve different kinds of problems related to rotational motion. Note that accurate and precise time measurement significantly impacts the reliability of other calculated variables. When rounding, as in the original exercise where the final answer was rounded to the nearest thousandth, it's crucial to maintain this precision for consistent results.