Problem 11 - Basic Technical Mathematics with Calculus

Question

From a point on the South Rim of the Grand Canyon, it is found that the angle of elevation of a point on the North Rim is \(1.2^{\circ} .\) If the horizontal distance between the points is \(9.8 \mathrm{mi},\) how much higher is the point on the North Rim? Solve the given problems. Sketch an appropriate figure, unless the figure is given.

Step-by-Step Solution

Answer

The North Rim is approximately 1084 feet higher than the South Rim.

Step 1: Understand the problem

We need to find how much higher a point on the North Rim of the Grand Canyon is compared to a point on the South Rim. We know the horizontal distance between them and the angle of elevation from the South Rim to the North Rim.

Step 2: Sketch the scenario

Imagine a right triangle where the horizontal distance between the South Rim and the North Rim is the base, the height difference between the points is the height, and the line of sight (which forms the hypotenuse) makes an angle of elevation of 1.2 degrees with the base.

Step 3: Set up the tangent ratio

In a right triangle, the tangent of an angle is the ratio of the opposite side (height difference) to the adjacent side (horizontal distance). We write this as: \[ \tan(1.2^{\circ}) = \frac{\text{height}}{9.8} \]

Step 4: Solve for height

Rearrange the equation to solve for height: \[ \text{height} = 9.8 \times \tan(1.2^{\circ}) \] Calculate the value of \( \tan(1.2^{\circ}) \) and then multiply by 9.8.

Step 5: Calculate the result

Compute \( \tan(1.2^{\circ}) \), which is approximately 0.02095. Then multiply it by 9.8: \[ \text{height} = 9.8 \times 0.02095 = 0.20531 \text{ mi} \] Convert miles to feet since the height difference is more commonly measured in feet (1 mile = 5280 feet): \[ 0.20531 \text{ mi} \times 5280 \text{ ft/mi} = 1084.00 \text{ ft} \]

Step 6: Conclusion

The point on the North Rim is approximately 1084 feet higher than the point on the South Rim.

Key Concepts

Angle of Elevation

The angle of elevation is a crucial concept in trigonometry, especially when dealing with heights and distances. It is the angle formed between the horizontal line of sight and the line of sight up to an object. This is relevant when you look up at something taller than your current level, like a mountain peak or a building.

In our scenario with the South and North Rims of the Grand Canyon, the angle of elevation is given as \(1.2^{\circ}\). This means from a point on the South Rim, if you were to look straight ahead horizontally and then up towards the top of the North Rim, you would elevate your line of sight by \(1.2^{\circ}\) to visualize it. The angle of elevation is always measured with respect to the horizontal.

To understand this better, imagine using a protractor. You measure the angle upwards from the horizontal level towards the peak. It's that specific upward angle that we're focusing on here. This is important because it helps in calculations related to height differences, as we use trigonometric functions based on this angle.

Tangent Function

The tangent function is a trigonometric function that relates the angle of a right triangle to the ratio of the opposite side to the adjacent side. In simpler terms, for a right angle triangle:

• The tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side.

In mathematical terms, for an angle \( \theta \), this can be expressed as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

In the exercise, the angle \(1.2^{\circ}\) is our \( \theta \), the height difference we're trying to find is the opposite side, and the horizontal distance \((9.8 \text{ mi})\) is the adjacent side. Therefore, to find the height difference, we set up the following equation using the tangent function:
\[ \tan(1.2^{\circ}) = \frac{\text{height}}{9.8} \]

This allows us to solve directly for the height by rearranging the equation.

Right Triangle

Understanding a right triangle is vital in problems involving trigonometry as it simplifies the calculation process using trigonometric functions. A right triangle has one angle of exactly 90 degrees, and this structure enables us to apply trigonometric ratios like sine, cosine, and tangent effectively.

In the context of calculating the height difference between the South and North Rims of the Grand Canyon, we can visualize a right triangle. Here's how it looks:

• The base of the triangle is the straight-line distance of 9.8 miles between the two points.
• The vertical side, or height, is what we're aiming to calculate. This represents how much higher the North Rim is compared to the South Rim.
• The hypotenuse is the line of sight from the South Rim to the North Rim, intersecting our line of elevation.

This setup allows us to use trigonometric functions to determine the unknown side, namely the height difference.

Height Calculation

After understanding the structure of the triangle and the principles behind the tangent function, calculating the height becomes straightforward. Here's a step-by-step outline:

• First, identify the values needed for your calculation. Here, you have the tangent of the angle (\( \tan(1.2^{\circ}) \)) and the horizontal distance (\(9.8 \text{ mi}\)).
• Next, calculate the tangent of \(1.2^{\circ}\). Using a calculator, find this to be approximately \(0.02095\).
• Use the tan function in the equation\ \[ \tan(1.2^{\circ}) = \frac{\text{height}}{9.8} \].Now solve for the height:
• Multiply \(9.8\) by \(0.02095\) to get the height in miles. This will give you \(0.20531 \text{ mi}\).
• Finally, convert this height from miles to feet for a more practical understanding. Since \(1 \text{ mi} = 5280 \text{ ft}\), multiply \(0.20531 \text{ mi}\) by \(5280 \text{ ft/mi}\), resulting in approximately \(1084 \text{ ft}\).

Thus, the North Rim is approximately 1084 feet higher than the South Rim. This method is very effective for such real-world applications and ensures accurate results from basic trigonometric principles.