Problem 52 - Contemporary Precalculus

Question

Determine the positive radian measure of the angle that the second hand of a clock traces out in the given time. 50 seconds

Step-by-Step Solution

Answer

Answer: Approximately 5.2359 radians.

Step 1: Determine the proportion of the circle completed by the second hand in 50 seconds.

There are 60 seconds in a minute, and in that time, the second hand completes one full circle, or 360 degrees. To find the proportion of the circle it has completed after 50 seconds, we can divide the 50 seconds by 60 seconds: Proportion = 50 seconds / 60 seconds

Step 2: Calculate the proportion as a decimal.

Converting the proportion calculated in Step 1 to a decimal gives us: Proportion = 50/60 = (5/6) = 0.8333 (rounded to four decimal places)

Step 3: Calculate the angle in degrees.

Now that we have the proportion as a decimal, we can multiply it by the number of degrees in a full circle to find the angle in degrees: Angle (degrees) = Proportion * 360 degrees = 0.8333 * 360 = 299.99 degrees (rounded to two decimal places)

Step 4: Convert the angle from degrees to radians.

To convert an angle from degrees to radians, we use the conversion factor (π radians / 180 degrees): Angle (radians) = Angle (degrees) * (π radians / 180 degrees) = 299.99 * (π/180) ≈ 5.2359 radians (rounded to four decimal places) After 50 seconds, the second hand of the clock traces out an angle of approximately 5.2359 radians.

Key Concepts

Angle conversion

Converting angles between degrees and radians is a fundamental skill in geometry and trigonometry. Radians and degrees are both units used to measure angles, but they represent different methods of measurement. In a full circle, there are 360 degrees, or equivalently, \(2\pi\) radians.
This connection is crucial because it allows us to convert between these two units easily. The conversion formula is based on this relationship:

• To convert from degrees to radians, use: \(\text{radians} = \text{degrees} \times \left(\frac{\pi}{180}\right)\)
• To go from radians to degrees, use: \(\text{degrees} = \text{radians} \times \left(\frac{180}{\pi}\right)\).

A situation may call for converting angle measurements, especially in physics or when dealing with periodic patterns like those found in clock movement.
For instance, if an angle is given in degrees, you may need the radian measure to understand its proportion relative to a circular path or solve a problem involving trigonometric functions.

Proportional reasoning

Proportional reasoning involves understanding the relationship between different quantities and using that relationship to make calculations. It's especially useful when dealing with circular motions or other periodic phenomena.
In our example of a clock's second hand, we reason proportionally to find out how much of the circle is traced in a given time.

• The second hand completes a full circle, or 360 degrees, every 60 seconds.
• To find how far it travels in 50 seconds, we compare 50 seconds to the whole 60 seconds.

Mathematically, the fraction of the circle covered is given by \(\frac{50}{60}\), which yields approximately 0.8333 when converted to a decimal.
This decimal is crucial because it represents the fraction of the circle completed in those 50 seconds, leading us to calculate the corresponding angle in degrees before converting it to radians.

Circle geometry

Circle geometry involves the fundamental properties and measurements associated with a circle. A circle is a perfect geometric shape with several noteworthy characteristics. Understanding these can help solve problems related to angles and rotations, like the one we have with the clock's second hand.

• A circle has 360 degrees or \(2\pi\) radians.
• Every point is equidistant from the center, making a rotation around the circle a consistent motion.
• The concept of a unit circle, which is a circle with a radius of one, is crucial for trigonometric calculations.

Texts usually describe angle movement in circles by their relationship to radius and circumference, solidifying concepts such as angular displacement and arc length.
For example, when dealing with a clock, knowing that each full rotation represents one full circle helps in calculating how much of the circle the hand covers over time, important when finding angles in radians or degrees.