To find the height of a building given the angle of elevation of the sun and the length of the shadow, we can use trigonometry, specifically the tangent function. Here’s a step-by-step solution:
1. Understanding the problem: We are given the angle of elevation of the sun, [tex]\(67.8^\circ\)[/tex], and the length of the shadow it casts, 67.5 feet. We need to find the height of the building.
2. Recall trigonometric relationship: The tangent of an angle in a right triangle is the ratio of the length of the opposite side (height of the building) to the length of the adjacent side (length of the shadow). Mathematically, this is expressed as:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Where:
[tex]\[
\theta = 67.8^\circ
\][/tex]
[tex]\[
\text{opposite} = \text{height of the building}
\][/tex]
[tex]\[
\text{adjacent} = 67.5 \text{ feet}
\][/tex]
3. Set up the equation: Using the tangent function:
[tex]\[
\tan(67.8^\circ) = \frac{\text{height of the building}}{67.5}
\][/tex]
4. Solve for the height of the building:
[tex]\[
\text{height of the building} = \tan(67.8^\circ) \times 67.5
\][/tex]
5. Calculate the value of [tex]\(\tan(67.8^\circ)\)[/tex]: Using a calculator, find the tangent of [tex]\(67.8^\circ\)[/tex].
6. Perform the multiplication:
[tex]\[
\text{height of the building} \approx 2.450369879 \times 67.5 \approx 165.40370085747472
\][/tex]
7. Round the result: Round the height to one decimal place:
[tex]\[
\text{height of the building} \approx 165.4
\][/tex]
Thus, the height of the building is [tex]\(\boxed{165.4}\)[/tex] feet.
