Math Problem Statement

While sailing toward a​ statue, a sailor in a boat observed that at a certain​ point, the angle of elevation of the tip of the torch was 27 degrees. After sailing another 100 meters toward the​ statue, the angle of elevation became 38 degrees 50 prime.

HOW TALL IS THE STATUE?

(do not round until the final answer, then round to the nearest integer)

Solution

To determine the height of the statue, we can use trigonometry and the concept of right triangles. Here's the approach:

Step 1: Define variables and diagram the situation.

  1. Let hh be the height of the statue.
  2. Let dd be the distance from the sailor's initial position to the base of the statue.
  3. After sailing 100 meters toward the statue, the new distance becomes d100d - 100.

We now have two right triangles:

  • The first triangle has an angle of elevation of 2727^\circ and distance dd from the statue.
  • The second triangle has an angle of elevation of 385038^\circ 50' (which is 38+5060=38.833338 + \frac{50}{60} = 38.8333^\circ) and distance d100d - 100.

Step 2: Apply the tangent function.

For the first triangle: tan(27)=hd\tan(27^\circ) = \frac{h}{d}

For the second triangle: tan(38.8333)=hd100\tan(38.8333^\circ) = \frac{h}{d - 100}

Step 3: Solve the system of equations.

From the first equation, solve for hh: h=dtan(27)h = d \cdot \tan(27^\circ)

From the second equation, solve for hh as well: h=(d100)tan(38.8333)h = (d - 100) \cdot \tan(38.8333^\circ)

Now, set the two expressions for hh equal to each other: dtan(27)=(d100)tan(38.8333)d \cdot \tan(27^\circ) = (d - 100) \cdot \tan(38.8333^\circ)

Step 4: Solve for dd.

Expanding the right side: dtan(27)=dtan(38.8333)100tan(38.8333)d \cdot \tan(27^\circ) = d \cdot \tan(38.8333^\circ) - 100 \cdot \tan(38.8333^\circ)

Now, collect the dd-terms on one side: d(tan(27)tan(38.8333))=100tan(38.8333)d \cdot (\tan(27^\circ) - \tan(38.8333^\circ)) = -100 \cdot \tan(38.8333^\circ)

Solve for dd: d=100tan(38.8333)tan(27)tan(38.8333)d = \frac{-100 \cdot \tan(38.8333^\circ)}{\tan(27^\circ) - \tan(38.8333^\circ)}

Step 5: Compute the value of dd.

We can now compute the tangent values and solve for dd: tan(27)0.5095,tan(38.8333)0.8011\tan(27^\circ) \approx 0.5095, \quad \tan(38.8333^\circ) \approx 0.8011

Substitute these values into the equation: d=1000.80110.50950.8011d = \frac{-100 \cdot 0.8011}{0.5095 - 0.8011} d=80.110.2916274.81metersd = \frac{-80.11}{-0.2916} \approx 274.81 \, \text{meters}

Step 6: Calculate the height hh.

Now that we have dd, use the first equation to find hh: h=dtan(27)=274.810.5095140metersh = d \cdot \tan(27^\circ) = 274.81 \cdot 0.5095 \approx 140 \, \text{meters}

Final Answer:

The height of the statue is approximately 140 meters.

Would you like further details or clarifications? Here are five related questions:

  1. What are other real-life applications of trigonometry in navigation?
  2. How does changing the angle of elevation affect the height calculation?
  3. What other methods could be used to calculate heights indirectly?
  4. How would this problem change if the second angle was measured farther from the statue?
  5. What role does approximation play in solving trigonometric problems?

Tip: When dealing with trigonometry problems, always try to sketch a diagram to help visualize the situation.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangle
Angle of Elevation

Formulas

tan(θ) = opposite / adjacent
h = d * tan(θ)
d = -100 * tan(θ2) / (tan(θ1) - tan(θ2))

Theorems

Tangent Function in Right Triangles

Suitable Grade Level

Grades 10-12