A builder has to surmount a rise in elevation from 1260 ft to 1930 ft with a straight road.

Question

He has only been granted funding to build a mile long section of road, and regulations state that the road can't have a slope of more than 1/8. Can he complete his task? If so, give the horizontal distance from the start to the end of the road to the nearest inch. If not, give the slope to the nearest thousandth that would allow him to build the road. 

Answer 1

The change in elevation is 1930 - 1260 = 670 ft. This would be the vertical leg of a right triangle, or the rise of the stretch of road. If the slope of the road is not to exceed 1/8, then the run (horizontal change) would be 670×8 = 5360 ft. Using the Pythagorean Theorem, the hypotenuse (or the length of the road) would be d, where

 

d² = 5360² + 670² = 29178500

 

d = √29178500 ≈ 5401.71 ft.

 

What does this tell you? It says that if the slope is the largest allowed, the road would have to be longer than one mile (5280 feet). The next question is: what if the slope is less than 1/8?

 

You should think about whether decreasing the slope will shorten or lengthen the length of the road. Here is a question to get you started: What if the slope were 1/9 (which is a little smaller than 1/8)?

 

If you experiment like this, you will be able to answer the question.