Question

In Problems 55-57, use the facts that the radius of Earth is 3960 miles and 1 mile \(=5280\) feet. The conning tower of the U.S.S. Silversides, a World War II submarine now permanently stationed in Muskegon, Michigan, is approximately 20 feet above sea level. How far can you see from the conning tower?

Short answer

Approximately 3.87 miles.

Step-by-step solution

Convert Units

Convert the height of the conning tower from feet to miles. Since 1 mile = 5280 feet, \[ \text{Height in miles} = \frac{20}{5280} \text{ miles} \]

Calculate Earth's Radius Plus Height of Tower

Add the height of the conning tower (in miles) to the radius of the Earth: \[ R + h = 3960 + \frac{20}{5280} \text{ miles} \]

Apply the Distance Formula

To find how far the horizon appears from the conning tower, use the formula: ewline \[ d = \sqrt{2Rh + h^2} \] ewline where \( R \) is Earth’s radius (3960 miles) and \( h \) is the height of the conning tower (in miles). Note that \( h^2 \) is negligible because \( h \) is very small compared to \( R \). So, the formula simplifies to: \[ d \approx \sqrt{2Rh} \]

Substitute the Values

Substitute the values into the simplified formula: ewline \[ d \approx \sqrt{2 \times 3960 \times \frac{20}{5280}} \] ewline Simplify the value inside the square root: ewline \[ d \approx \sqrt{15} \approx 3.87 \text{ miles} \]

Key concepts

Earth's radius

To understand how far you can see from a specific height above sea level, you need to know the radius of the Earth. The Earth's radius is the distance from the center of the Earth to its surface. It is approximately 3960 miles. This large radius is important because it impacts how far the horizon is from any given height. When you stand on an elevated point, the curvature of the Earth affects the distance your eyes can reach. Knowing the Earth's radius helps us use mathematical formulas to calculate that distance.

unit conversion

In many problems, you need to convert units to make calculations easier. For example, in this problem, the height of the conning tower is given in feet, but the radius of the Earth is in miles. To make calculations simpler, converting the height into miles is necessary. Remember, 1 mile equals 5280 feet. So, to convert from feet to miles, you divide the number of feet by 5280. In our example:

• Height in feet: 20 feet
• Height in miles = 20 / 5280
• Height in miles = approximately 0.00379 miles
Converting units ensures consistency and accuracy in your calculations.

distance formula

To determine how far you can see from the conning tower, we use a formula that considers the Earth's radius and the height of the tower. The distance to the horizon can be found using: \[ d = \sqrt{2Rh + h^2} \] In this formula, \( R \) is the Earth's radius (3960 miles) and \( h \) is the height of the observation point (in miles). Since the height is very small in comparison to the Earth's radius, \( h^2 \) is negligible. This simplifies our formula to: \[ d \approx \sqrt{2Rh} \] Then, substituting the values for \( R \) and \( h \), we get: \[ d \approx \sqrt{2 \times 3960 \times 0.00379} \] Which simplifies to: \[ d \approx \sqrt{15} \approx 3.87 \text{ miles} \] This is the maximum distance you can see from the tower, considering the curvature of the Earth. Understanding this formula and the elements within it allows you to grasp the factors affecting visibility from different heights.