To find the rise in 6800 feet of horizontal run given a slope of [tex]\(2^\circ 40'\)[/tex], we can follow these steps:
1. Convert the slope angle from degrees and minutes to decimal degrees.
- Degrees: [tex]\(2^\circ\)[/tex]
- Minutes: [tex]\(40'\)[/tex]
We know that 1 degree is equal to 60 minutes. Therefore, to convert the minutes to degrees, we use the following conversion:
[tex]\[
\text{Decimal degrees} = 2 + \frac{40}{60} = 2 + 0.6667 = 2.6667 \, \text{degrees}
\][/tex]
2. Convert the slope angle from decimal degrees to radians.
Trigonometric functions typically use radians, so we need to convert the angle from degrees to radians using the formula:
[tex]\[
\text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right)
\][/tex]
Substituting the decimal degrees:
[tex]\[
\text{Radians} = 2.6667 \times \left( \frac{\pi}{180} \right) \approx 0.04654 \, \text{radians}
\][/tex]
3. Calculate the rise using the tangent function.
The rise can be determined using the tangent of the slope angle, which is the ratio of the rise to the horizontal run:
[tex]\[
\text{Tangent}(\theta) = \frac{\text{Rise}}{\text{Run}}
\][/tex]
Solving for rise:
[tex]\[
\text{Rise} = \text{Run} \times \text{Tangent}(\theta)
\][/tex]
Substituting the values for the run and the angle in radians:
[tex]\[
\text{Rise} = 6800 \times \tan(0.04654) \approx 316.7151 \, \text{feet}
\][/tex]
4. Round the result to the nearest foot.
Finally, rounding 316.7151 to the nearest foot:
[tex]\[
\text{Rise} \approx 317 \, \text{feet}
\][/tex]
Therefore, the rise in 6800 feet of horizontal run is approximately [tex]\(317\)[/tex] feet.
