Problem 12 - Algebra 2

Question

Earth rotates on its axis once every 24 hours. How long does it take Earth to rotate through an angle of \(\frac{\pi}{6} ?\)

Step-by-Step Solution

Answer

It takes Earth 2 hours to rotate \(\frac{\pi}{6}\) radians.

Step 1: Understand the Problem

Earth rotates 360 degrees, or a full circle, in 24 hours. We need to find out how long it takes to rotate through an angle of \(\frac{\pi}{6}\) radians.

Step 2: Convert Angle to Degrees

Since \(2\pi\) radians is equal to 360 degrees, \(\frac{\pi}{6}\) radians is equal to \(\frac{\pi}{6} \times \frac{360}{2\pi}\) degrees. Simplifying gives us \(\frac{\pi}{6} \times 180 = 30\) degrees.

Step 3: Proportion of Rotation

Calculate the proportion of rotation: \(\frac{30}{360}\). This simplifies to \(\frac{1}{12}\).

Step 4: Calculate Time for Rotation

Since Earth takes 24 hours to rotate 360 degrees, the time it takes to rotate \(30\) degrees (or \(\frac{\pi}{6}\) radians) is \(\frac{1}{12} \times 24\) hours, which equals 2 hours.

Key Concepts

Angle Conversion

Understanding how to convert between different units of angle measurement is a fundamental concept in many areas of math and science. Angles can be measured in degrees or radians, which are two of the most common units. A full circle is 360 degrees, but in radian measure, it is expressed as \(2\pi\) radians. This is because radians are defined based on the radius of a circle. To convert an angle from radians to degrees, use the formula:

• Degrees = Radians \(\times \frac{180}{\pi}\)

Applying this formula allows for easy switching between these units. This is particularly useful when dealing with problems involving the Earth's rotation or astronomy, where angles are often expressed in radians. Ensuring you have a good grasp of angle conversion will make solving related problems much more straightforward.

Radians to Degrees

Radians to degrees conversion is a critical process when working with angles in mathematics. It's often required because different applications prefer different units; for instance, degrees are sometimes easier to understand intuitively. In our exercise, converting \(\frac{\pi}{6}\) radians to degrees involves multiplying by \(\frac{180}{\pi}\). Thus,

• \(\frac{\pi}{6}\) radians \(\times \frac{180}{\pi} = 30\) degrees

These computations simplify our understanding, converting abstract radian measures into degrees, which we use regularly in everyday contexts. By practicing these conversions, you'll develop a keener intuition for interpreting and calculating angles.

Proportional Rotation

Proportional rotation is key for understanding how much an object, like Earth, rotates over a given period. Once an angle is converted to degrees, identifying the rotation proportion tells us what fraction of a full rotation (360 degrees) it represents. For example, an angle of 30 degrees constitutes

• \(\frac{30}{360} = \frac{1}{12}\)

of a full rotation. Recognizing this proportion helps in calculating time or motion for partial rotations. Understanding these fractions can be crucial for problems involving circular motion or periodic events, like Earth's rotation on its axis.

Calculation of Time

Understanding the time calculation for proportional rotations revolves around the principle that the entire 360-degree rotation takes a specific amount of time, in this scenario, 24 hours for Earth's full rotation. By knowing the proportional rotation, we can derive the time for this partial spin. For instance, if the Earth rotates through a

• \(\frac{1}{12}\) of its axis in 24 hours, the time to complete this portion is:
• \(\frac{1}{12} \times 24 = 2\) hours

This calculation is useful not just for understanding Earth's rotation but also in fields like engineering and physics, where motion and time relations are crucial. Practicing these calculations can greatly aid in honing your ability to solve similar dynamic problems.