To find the height of the Eiffel Tower given that Bert is 2215 feet away from its base and the angle of elevation to the top of the tower is 24 degrees, we can use some trigonometry, specifically the tangent function. Here's how we solve this problem step-by-step:
1. Understand the Problem:
- We are given:
- Distance from the base of the tower to Bert: 2215 feet (this will be our 'adjacent' side in the trigonometric function).
- Angle of elevation: 24 degrees.
2. Set Up the Trigonometric Relationship:
- The tangent of an angle in a right triangle is the ratio of the 'opposite' side to the 'adjacent' side:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
3. Identify Components:
- Here, [tex]\(\theta = 24^\circ\)[/tex].
- The 'adjacent' side is the distance from Bert to the base of the tower, which is 2215 feet.
- The 'opposite' side is the height of the Eiffel Tower that we need to find.
4. Formulate the Equation:
- Using the tangent function:
[tex]\[
\tan(24^\circ) = \frac{\text{height of the Eiffel Tower}}{2215}
\][/tex]
5. Solve for the Height:
- To isolate the height of the tower, multiply both sides of the equation by 2215:
[tex]\[
\text{height of the Eiffel Tower} = 2215 \times \tan(24^\circ)
\][/tex]
6. Calculate the Angle in Radians:
- Trigonometric functions in most calculators and programming languages often require angles to be in radians.
- Convert 24 degrees to radians:
[tex]\[
24^\circ = 0.4188790204786391 \text{ radians}
\][/tex]
7. Find the Tangent:
- Calculate [tex]\(\tan(0.4188790204786391)\)[/tex].
8. Multiply to Find the Height:
- Use the value of the tangent calculated above:
[tex]\[
\text{height of the Eiffel Tower} = 2215 \times 0.4452286853085361
= 986.1815379584077 \text{ feet}
\][/tex]
So, the height of the Eiffel Tower, based on Bert’s observation, is approximately [tex]\(986.18\)[/tex] feet.
