Join the growing community of curious minds on IDNLearn.com. Our experts are ready to provide prompt and detailed answers to any questions you may have.
Solve the problem.
From a boat on the river below a dam, the angle of elevation to the top of the dam is [tex][tex]$12^{\circ} 4^{\prime}$[/tex][/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)?
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.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! For precise answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.
