Certainly! Let's solve this trigonometry problem step-by-step.
### Problem:
From a boat on the river below a dam, the angle of elevation to the top of the dam is [tex]\(12^\circ 4'\)[/tex]. If the dam is 912 feet above the level of the river, how far is the boat from the base of the dam (to the nearest foot)?
### Solution:
1. Given Information:
- Height of the dam, [tex]\( h \)[/tex] = 912 feet
- Angle of elevation, [tex]\(\theta\)[/tex] = [tex]\(12^\circ 4'\)[/tex]
2. Convert the angle to decimal degrees:
Angles are often given in degrees and minutes. To convert from degrees and minutes to decimal degrees:
[tex]\[
12^\circ 4' = 12 + \frac{4}{60} = 12^\circ + 0.0666667^\circ = 12.0666667^\circ
\][/tex]
3. Identify the right triangle formed:
- The height of the dam ([tex]\( h \)[/tex]) corresponds to the opposite side of the angle of elevation.
- The distance from the boat to the base of the dam corresponds to the adjacent side ([tex]\( d \)[/tex]).
- The angle at the boat is the angle of elevation [tex]\(\theta\)[/tex].
4. Use the tangent function:
The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d}
\][/tex]
Rearrange the equation to solve for [tex]\( d \)[/tex]:
[tex]\[
d = \frac{h}{\tan(\theta)}
\][/tex]
5. Plug in the values:
[tex]\[
d = \frac{912}{\tan(12.0666667^\circ)}
\][/tex]
6. Calculate the distance:
Using a calculator to find:
[tex]\[
\tan(12.0666667^\circ) \approx 0.213592
\][/tex]
Now, divide the height by the tangent of the angle:
[tex]\[
d = \frac{912}{0.213592} \approx 4266.21
\][/tex]
7. Round to the nearest foot:
The distance from the boat to the base of the dam is approximately 4266 feet.
### Conclusion:
The boat is approximately 4266 feet from the base of the dam.
