Problem 54 - Precalculus: Functions and Graphs

Question

Magnetic pole drift the geographic and magnetic north poles have different locations. Currently, the magnetic north pole is drifting westward through 0.0017 radian per year, where the angle of drift has its vertex at the center of Earth. If this movement continues, approximately how many years will it take for the magnetic north pole to drift a total of \(5^{\circ} 7\)

Step-by-Step Solution

Answer

It will take approximately 53 years.

Step 1: Convert Degrees to Radians

First, convert the total drift in degrees to radians. The formula for converting degrees to radians is \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). Here, the total drift is \( 5^{\circ} 7' \). Convert \( 7' \) (minutes) to degrees by noting that \( 1^{\circ} = 60' \), so: \( 7' = \frac{7}{60} \) degrees.The total in degrees: \( 5 + \frac{7}{60} = 5.1167^{\circ} \).Now convert degrees to radians: \[ 5.1167 \times \frac{\pi}{180} \approx 0.0893 \text{ radians} \]

Step 2: Calculate the Number of Years for Total Drift

Next, calculate how many years it will take for the magnetic north pole to drift the total radian measure found in Step 1. Given the drift rate is \(0.0017\) radians per year, use the formula \[ \text{Years} = \frac{\text{Total drift in radians}}{\text{Drift rate per year}} \].Substitute the values: \[ \text{Years} = \frac{0.0893}{0.0017} \approx 52.53 \]

Step 3: Round to Nearest Whole Year

Since time in years should be expressed as an integer, round 52.53 to the nearest whole number. Thus, \(52.53\) rounds to \(53\) years.

Key Concepts

Magnetic Pole Drift

Magnetic pole drift refers to the gradual movement of Earth's magnetic north pole across the globe. Unlike the geographic north pole, which is fixed, the magnetic north pole is constantly shifting due to changes in Earth's outer core where molten iron generates the magnetic field. This drift impacts navigation systems, compasses, and can even affect wildlife reliant on Earth's magnetic field for migration. Notably, the magnetic north pole is currently drifting at a rate of 0.0017 radians per year westward. Understanding this rate helps us predict when significant shifts might affect navigation and communication systems. This steady drift is slow but has a cumulative impact over time, requiring periodic adjustments to navigational charts and instruments.

Degrees to Radians Conversion

Converting degrees to radians is a crucial skill in precalculus, especially when dealing with angles, such as in the case of magnetic pole drift. Angles are often given in degrees, but in many mathematical and physical applications, radians are the preferred unit. To convert degrees into radians, use the formula:

• Radian measure = Degrees × (π/180)

This formula is derived from the fact that a complete circle is 360 degrees or 2π radians. Therefore, 1 degree is equivalent to π/180 radians. For smaller units, such as the 7 arcminutes in the exercise, you first convert them into degrees:

• 7' = 7/60 degrees

Adding these to the degree measure provides the total angle in degrees, which can then be converted to radians for further calculations.

Calculating Time Based on Rate

When calculating how long it will take for an event to occur given a constant rate, you divide the total amount of change needed by the rate of change. In the exercise regarding magnetic pole drift:

• The total change is the drift angle in radians.
• The rate is the drift per year.

The formula is:

• Time (years) = Total drift in radians / Drift rate per year

This calculation allows you to estimate how many years it will take for the magnetic pole to drift a specified amount, given its current rate. It's a straightforward way to apply rates to real-world problems.

Earth's Magnetic Field

Earth's magnetic field is a vast and complex force of nature that extends from Earth's interior out into space. It is primarily generated by the movement of molten iron and nickel in Earth's outer core. This magnetic field not only causes compass needles to point north but also shields the planet from harmful solar and cosmic radiation. On a broader scale, it plays a vital role in sustaining life on Earth by maintaining the atmosphere. Changes in Earth's magnetic field, such as pole drift, can have wide-ranging effects on both natural systems and human technology. Satellite, navigation, and communication systems are all influenced by changes and variations in the magnetic field, highlighting the importance of understanding and monitoring these magnetic phenomena.