To determine the height of the building, we can use trigonometry. Specifically, we will use the tangent function, which is commonly utilized to relate the angles in a right triangle to the lengths of the opposite and adjacent sides.
Here's a step-by-step breakdown:
1. Understand the problem:
- We are given an angle of elevation of the sun which is 67.8°.
- The shadow cast by the building is 67.5 feet long.
- We need to determine the height of the building.
2. Identify the relevant trigonometric function:
- In a right triangle, the tangent of an angle is defined as 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 written as:
[tex]\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
3. Set up the equation:
- Let [tex]\( h \)[/tex] be the height of the building.
- We are given the angle [tex]\(\theta = 67.8^\circ \)[/tex] and the length of the adjacent side (shadow length) is 67.5 feet.
- The equation using the tangent function will be:
[tex]\[
\tan(67.8^\circ) = \frac{h}{67.5}
\][/tex]
4. Solve for [tex]\( h \)[/tex]:
- Rearrange the equation to solve for [tex]\( h \)[/tex]:
[tex]\[
h = \tan(67.8^\circ) \times 67.5
\][/tex]
5. Calculate [tex]\( \tan(67.8^\circ) \)[/tex]:
- Using a calculator or trigonometric tables, find the tangent of 67.8 degrees.
6. Multiply the obtained value by the shadow length:
- Perform the multiplication:
[tex]\[
h \approx 2.449 \times 67.5 \approx 165.4
\][/tex]
7. Conclude:
- The height of the building is [tex]\( 165.4 \)[/tex] feet when rounded to one decimal place.
So, the height of the building is:
[tex]\[ 165.4 \][/tex]
