A straight trail with a uniform inclination of 17deg leads from a lodge to an elevation of 900 feet to a mountain lake at a elevation of 5500 ft. what is the length of the trail?
(im soo stuck on this problem)
let the length of the trail be t ft.
then t is the hypotenuse of a right-angled triangle of height 4600 ft
so sin 17 = 4600/t
t = 4600/sin17
= ... (don't have a calculator handy)
To find the length of the trail, we can use trigonometry and the concept of right triangles. Here's how you can solve the problem step by step:
Step 1: Identify the information given:
- The starting elevation of the lodge is 900 feet.
- The ending elevation of the mountain lake is 5500 feet.
- The trail has a uniform inclination of 17 degrees.
Step 2: Determine the vertical distance of the trail:
- The vertical distance is the difference in elevation between the starting and ending points: 5500 ft - 900 ft = 4600 ft.
Step 3: Set up a right triangle:
- In a right triangle, the vertical distance represents the opposite side, and the length of the trail represents the hypotenuse.
- We need to find the length of the trail, which is the hypotenuse.
Step 4: Apply trigonometry:
- The trigonometric function that relates the opposite side and the hypotenuse in a right triangle is the sine function.
- In this case, we can use the sine function to relate the vertical distance (4600 ft) with the angle of inclination (17 degrees).
- The sine of an angle is equal to the ratio of the opposite side to the hypotenuse: sin(angle) = opposite/hypotenuse.
Step 5: Solve for the length of the trail:
- Rearrange the equation to solve for the hypotenuse (length of the trail): hypotenuse = opposite / sin(angle).
- Plugging in the values: hypotenuse = 4600 ft / sin(17 degrees).
Step 6: Use a calculator to find the length of the trail:
- Use a scientific calculator or trigonometric table to calculate sin(17 degrees).
- Then divide 4600 ft by the value of sin(17 degrees) to get the length of the trail.
Following these steps, you should be able to calculate the length of the trail.