Problem 70 - Precalculus

Question

A city is located at 75 degrees north latitude. Assume the radius of the earth is 3960 miles and the earth rotates once every 24 hours. Find the linear speed of a person who resides in this city.

Step-by-Step Solution

Answer

The linear speed is approximately 268.37 miles per hour.

Step 1: Understand the Problem

To find the linear speed of a person residing in a city at 75 degrees north latitude due to Earth's rotation, we first need to identify the circular path traveled by the person. Since Earth is a sphere, its rotation causes every point on its surface to trace a circular path parallel to the equator, dependent on the latitude.

Step 2: Determine the Radius of the Circular Path

The radius of the circular path at a given latitude can be calculated by multiplying Earth's radius by the cosine of the latitude. Thus for 75 degrees latitude, the radius of the circular path is \[ r = 3960 \times \cos(75^\circ) \] Calculate this using a calculator.

Step 3: Calculate the Circumference of the Path

Once the radius of the path is known, calculate the circumference of the circular path using \[ C = 2\pi r \]where \( r \) is the radius found in Step 2.

Step 4: Determine the Time of One Rotation

Note that the Earth rotates once every 24 hours. Thus, the time for one complete rotation is 24 hours or \( 24 \times 3600 \) seconds.

Step 5: Calculate the Linear Speed

The linear speed \( v \) is given by the formula: \[ v = \frac{C}{T} \] where \( C \) is the circumference and \( T \) is the time for one rotation. Substitute the values obtained from Steps 3 and 4 to get the linear speed.

Step 6: Compute the Numerical Result

Calculate each step value explicitly:- First compute \( r = 3960 \times \cos(75^\circ) \approx 3960 \times 0.2588 \approx 1024.77 \) miles.- Then the circumference \( C = 2\pi \times 1024.77 \approx 6440.8 \) miles.- Finally, compute the linear speed \( v = \frac{6440.8}{24} \approx 268.37 \) miles per hour.

Key Concepts

Latitude

Latitude is a geographical term that denotes a specific location on the Earth's surface. It is measured in degrees, ranging from 0° at the Equator to 90° at the poles. Latitude determines how far north or south a place is relative to the Equator.
For instance, a city at 75 degrees north latitude, like the one in our exercise, is quite far from the Equator and is located closer to the North Pole. This position affects the circular path the inhabitants travel due to Earth's rotation.

• Latitudes vary from 0° to 90° north and south.
• The Equator represents 0° latitude, where the Latitude angle is the smallest.

As you move away from the Equator towards the poles, the linear distance between degrees decreases. This results in smaller circular paths for higher latitudes, affecting linear speed.

Earth's Rotation

The Earth rotates around its axis once every 24 hours. This rotation causes continuous day and night cycles and affects everything on Earth, from weather patterns to human activity.
The circular motion created by this rotation makes any point on the Earth’s surface move at a certain linear speed, dependent on its location or latitude.

• Earth completes one full rotation in approximately 24 hours.
• This steady rotation is what gives us regular sunrise and sunset.
• Every point on the Earth moves in a circle once a day due to this rotation.

Since Earth rotates from west to east, everything on it travels that way, with linear speed varying by latitude. More northern or southern locations move slower compared to those near the Equator because they trace smaller circles.

Circular Path

A circular path refers to the path traced by a point on Earth's surface as the planet rotates. Due to Earth's spherical shape, every point on the surface moves circularly about the axis.
The size of this circular path depends heavily on the latitude.

• The circle is largest at the Equator and decreases in size toward the poles.
• The radius of the circular path is affected by the Earth's radius and latitude.

At 75 degrees north latitude, the circular path is smaller because the point on the surface is closer to the axis. To calculate the radius of this path, you multiply the Earth's radius by the cosine of the latitude angle. This calculated radius then helps you find the exact circumference and, subsequently, the linear speed.

Radius of Earth

The radius of Earth is a crucial component in calculating the circular path traveled due to Earth’s rotation. It is assumed to be approximately 3960 miles. This value is critical because it helps determine the radius of the actual path at any given latitude.
The radius of the Earth serves as the baseline from which other distances and calculations branch out, enabling you to determine how far someone travels in a day due to Earth's rotation.

• Earth’s radius is essential for calculating distances in planetary measurements.
• This constant helps translate angles (latitude) into tangible, linear distances on the surface.

By multiplying Earth’s radius by the cosine of the latitude angle, you find the radius of the circular path specific to that location, crucial for understanding how fast the surface moves in relation to Earth’s rotation.