Sure, let's solve this problem step-by-step!
Question:
A point on the ground is 50 feet from my house. The angle of elevation to the top of the house is 48°. Find the height of the house to the nearest tenth.
Solution:
1. Understand the Problem:
- You are given a right triangle where:
- The distance from the point on the ground to the base of the house (adjacent side) is 50 feet.
- The angle of elevation from this point to the top of the house is 48°.
- You need to find the height of the house (opposite side).
2. Draw a Diagram:
- Draw a right triangle.
- Label the angle of elevation (θ) as 48°.
- Label the adjacent side (distance from the point to the house) as 50 feet.
- Label the opposite side (height of the house) as [tex]\( h \)[/tex], which we need to find.
3. Use Trigonometry:
- In a right triangle, the tangent function relates the opposite side (height of the house) to the adjacent side (distance to the house).
The formula is:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- Substitute the given values into the formula:
[tex]\[
\tan(48°) = \frac{h}{50}
\][/tex]
4. Calculate [tex]\( h \)[/tex]:
- Rearrange the formula to solve for [tex]\( h \)[/tex]:
[tex]\[
h = 50 \times \tan(48°)
\][/tex]
5. Compute Tangent of 48°:
- Using the tangent value of 48° from trigonometric tables or a calculator, we get:
[tex]\[
\tan(48°) \approx 1.1106
\][/tex]
6. Multiply to Find Height:
- Now, calculate the height:
[tex]\[
h = 50 \times 1.1106 \approx 55.5306
\][/tex]
7. Round the Result:
- Finally, round the height to the nearest tenth:
[tex]\[
h \approx 55.5
\][/tex]
Therefore, the height of the house is approximately 55.5 feet.
