Problem 41 - Thinking Mathematically

Question

The Statue of Liberty is approximately 305 feet tall. If the angle of elevation of a ship to the top of the statue is \(23.7^{\circ}\), how far, to the nearest foot, is the ship from the statue's base?

Step-by-Step Solution

Answer

After doing the calculations, the ship is approximately 746 feet away from the base of the Statue of Liberty.

Step 1: Convert Degrees to Radians

First, we need to convert the given angle from degrees to radians. We can do this by multiplying the given degree by \(\frac{\pi}{180}\). So, \(23.7^{\circ} = 23.7 \times \frac{\pi}{180}\) radians.

Step 2: Apply the Tangent Formula

Next, we use the formula for tangent of an angle, which is `opposite/adjacent`. We rearrange the formula to find the 'adjacent' side, so `adjacent = opposite/tan(angle)`. In this problem, the 'opposite' side is the height of the Statue of Liberty which is 305 feet. So, `adjacent = 305 / tan(23.7)`.

Step 3: Calculate the Distance

Finally, we calculate the value of the above expression to find the distance. To round off to the nearest foot, if the decimal part is less than 0.5, then we take the lower nearest value, but if the decimal part is 0.5 or more, we take the upper nearest value.

Key Concepts

Angle of Elevation

When observing an object from a lower point, the "angle of elevation" comes into play. It is the angle between the horizontal line of sight and the line that stretches up to the object. This concept is crucial in trigonometry.

• Imagine you're looking at a tall building from afar. As you tilt your head up to see the top, you are observing an angle of elevation.
• This angle helps in solving problems related to heights and distances.

The angle of elevation is key in determining how far an object is, by considering this tilt from your line of sight.

Tangent Formula

The "tangent formula" is a vital tool in trigonometry for finding distances. It is defined as the ratio of the opposite side to the adjacent side in a right triangle.To use the tangent formula effectively:

• Identify the angle and determine which sides of the triangle are opposite and adjacent to it.
• The formula is: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• By rearranging, you can solve for the unknown side: \( \text{adjacent} = \frac{\text{opposite}}{\tan(\theta)} \).

This is how we calculate the horizontal distance (adjacent side) from an object when its height (opposite side) and angle of elevation are known.

Degrees to Radians Conversion

Angles can be measured in degrees or radians. In trigonometry, you'll often need to convert from one to the other. The conversion factor is \( \pi/180 \).Here's the process:

• To convert degrees to radians, multiply the degree measure by \( \frac{\pi}{180} \).
• For example, to convert an angle of \( 23.7^{\circ} \) to radians, compute \( 23.7 \times \frac{\pi}{180} \).
• Radians are often used in mathematical calculations because they simplify terms.

Converting to radians ensures that trigonometric functions work properly since they typically operate in radians.

Distance Calculation

"Distance calculation" in trigonometry often involves applying trigonometric formulas to determine lengths of sides in a triangle.When calculating distance to an object like the Statue of Liberty:

• Use the tangent formula: \( \text{adjacent} = \frac{\text{opposite}}{\tan(\text{angle})} \).
• Here, the opposite side is known (e.g., the height of the object). Calculate the tangent of the angle using a calculator set to radians.
• Solve to find the adjacent side, which is the horizontal distance from the observer to the object's base.

After calculating, remember to round off the distance to the nearest unit. This provides a practical and precise measurement for real-world situations. Always confirm that you understand each step involved by verifying calculations.