Question

A railroad track has an angle of elevation of 1.00°. What is the difference in height (in ft) of two points on the track 9.00 mi apart? (1 mi = 5,280 ft. Round your answer to three significant digits.)

Answer 1

To determine the height difference, convert the distance to feet, then use the tangent of the angle of elevation and horizontal distance to find the height change, which is approximately 830 feet after rounding.

Step-by-step explanation:

To solve the mathematical problem completely, we should use trigonometric concepts to find the difference in height between two points on a railroad track with a 1.00° angle of elevation 9.00 miles apart. First, we need to convert the distance from miles to feet: 9 miles × 5,280 feet per mile = 47,520 feet.

Now, considering the track as the hypotenuse of a right triangle where the angle of elevation is 1.00°, we can use the tangent function, which is equal to the opposite side (height difference) over the adjacent side (horizontal distance). So, tan(1.00°) = Height Difference / 47,520 feet.

To find the height difference, we multiply both sides by 47,520 feet, getting Height Difference = 47,520 feet × tan(1.00°). Calculating this, we get Height Difference = 47,520 feet × 0.017455 = approximately 829.85 feet. However, when rounding to three significant digits, the difference in height is 830 feet.