To determine the height of the house given that a point on the ground is 50 feet from the house and the angle of elevation to the top of the house is [tex]\(48^\circ\)[/tex], follow these steps:
1. Understand the Problem:
- Distance from the house (adjacent side in a right triangle): [tex]\(50\)[/tex] feet
- Angle of elevation: [tex]\(48^\circ\)[/tex]
- We need to find the height of the house (opposite side in a right triangle).
2. Convert the Angle of Elevation to Radians:
The angle of elevation provided is in degrees and we often need to work with radians in trigonometric calculations.
[tex]\[
\text{Angle in radians} = 0.8377580409572782
\][/tex]
3. Use the Tangent Function:
Tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
[tex]\[
\tan(48^\circ) = \frac{\text{Opposite side (height of the house)}}{\text{Adjacent side (distance from the house)}}
\][/tex]
4. Set Up the Equation and Solve for the Height:
Given:
[tex]\[
\tan(48^\circ) = \frac{\text{height of the house}}{50}
\][/tex]
We can express the height of the house as:
[tex]\[
\text{Height of the house} = 50 \times \tan(48^\circ)
\][/tex]
5. Calculate the Height:
Using the known value:
[tex]\[
\tan(48^\circ) = 1.1106
\][/tex]
Thus:
[tex]\[
\text{Height of the house} = 50 \times 1.1106 = 55.53062574145965 \text{ feet}
\][/tex]
6. Round the Height to the Nearest Tenth:
Finally, we round the height to the nearest tenth:
[tex]\[
\text{Height of the house} \approx 55.5 \text{ feet}
\][/tex]
Therefore, the height of the house is approximately [tex]\(55.5\)[/tex] feet when rounded to the nearest tenth.
