Problem 44 - Basic Technical Mathematics with Calculus

Question

The CN Tower in Toronto has an observation deck at \(346 \mathrm{m}\) above the ground. Assuming ground level and Lake Ontario level are equal, how far can a person see from the deck? (The radius of Earth is 6378 km.)

Step-by-Step Solution

Answer

A person can see approximately 66.42 km from the observation deck.

Step 1: Identify Variables

First, identify the known variables: the height of the observation deck is \(h = 346\, \text{m}\) or \(0.346\, \text{km}\) and the radius of the Earth is \(R = 6378\, \text{km}\). We aim to find the distance \(d\), which is how far a person can see from this height.

Step 2: Use the Line of Sight Formula

The line of sight distance can be calculated using the formula \(d = \sqrt{2Rh + h^2}\). This formula comes from considering the Earth as a circle and forming a right triangle with the radius and height.

Step 3: Substitute Known Values

Substitute the known values into the formula: \[d = \sqrt{2 \times 6378 \times 0.346 + (0.346)^2}\].

Step 4: Calculate the Intermediate Values

Calculate the multiplication inside the square root: \(2 \times 6378 \times 0.346 = 4414.728\) and \(0.346^2 = 0.119716\).

Step 5: Sum the Values

Add the two intermediate results: \(4414.728 + 0.119716 = 4414.847716\).

Step 6: Compute the Square Root

Finally, compute the square root of the sum to find \(d\): \(d = \sqrt{4414.847716} \approx 66.42\, \text{km}\).

Key Concepts

The Right Triangle

The right triangle is a key element in understanding many concepts in trigonometry, including the line of sight from a point above a sphere, such as the Earth. In a right triangle, one of the angles is exactly 90 degrees. This unique feature allows us to use various trigonometric functions and properties to solve problems. Consider the right triangle formed by the center of the Earth (one vertex), the viewer on the observation deck (another vertex), and the point on the horizon that is visible from the deck (the final vertex). The side extending from the Earth’s center to the horizon is the Earth’s radius, while the side from the Earth’s center to the observer is the sum of the Earth’s radius and the height of the observation deck. The line of sight from the observer to the horizon completes the triangle.

• The hypotenuse is the sum of the Earth’s radius and the observation deck's height.
• One leg of the triangle is the Earth's radius.
• The other leg is the line of sight, the distance the problem asks us to calculate.

Utilizing this configuration allows us to apply the Pythagorean Theorem in scenarios beyond flat surfaces, such as the Earth's curved surface.

Line of Sight Formula

The line of sight formula is crucial for calculating how far one can see when at an elevated position above a curved surface, such as the Earth. The formula for the distance one can see is given by: \[ \text{Distance} = \sqrt{2Rh + h^2} \]Where:

• \(R\) is the Earth's radius (in our case, 6378 km).
• \(h\) is the height of the observer above Earth's surface (346 meters, converted to kilometers is 0.346 km).

The derivation of this formula relies on the principles of geometry involving a right triangle. It simplifies the problem to a manageable mathematical equation, emphasizing how a simple change in perspective or height can allow us to see over significant distances. This formula helps convert the conceptual challenge of viewing range into a practical calculation, harnessing both algebra and geometry.

Distance Calculation

Calculating the distance one can see involves using the line of sight formula explained previously. We start by plugging the known values into the formula:\[d = \sqrt{2 \times 6378 \times 0.346 + (0.346)^2} \]First, calculate the intermediate product and square:

• The product of the Earth's circumference and the height: \(2 \times 6378 \times 0.346 = 4414.728\).
• The square of the height: \( (0.346)^2 = 0.119716\).

Add these results to get a cumulative value inside the square root: \[ 4414.728 + 0.119716 = 4414.847716 \]Complete the calculation by taking the square root, yielding approximately 66.42 kilometers. This calculated distance indicates how far one can see from the observation deck when looking straight ahead at the horizon on a clear day. It's worth noting that actual visibility can vary with atmospheric conditions and obstructions, which is one of the real-world considerations beyond mathematical calculations.