If you drive 0.6 miles along the road and your altitude increases by 150 feet, what is the angle of inclination of the road? Give your answer to two decimal places. Note that one mile is equal to 5280 feet.
It can be really useful to sketch a diagram in these sorts of scenarios. It will allow you to identify the type of question it is and what you will need to use to be able to solve it.
Here we have a right-angled triangle, with the hypotenuse representing the slope of the road and the adjacent to the angle representing the horizontal. Currently, we have a measurement for the slope as 0.6 miles and a height of 150 feet.
Before we can go any further, we must ensure all measurements have the same units. We’re given that one mile is equal to 5280 feet. We can therefore multiply 0.6 by 5280 to convert 0.6 miles into 3168 feet.
Now we have a right-angled triangle with two sides given, for which we need to find an angle. In that case, we need to use right angle trigonometry to find the missing angle. For 𝜃, which represents the angle of the inclination of the road, we currently know the length of both the hypotenuse and the opposite. In this case, we therefore need to use the sine ratio.
Substituting what we know into our formula for sin 𝜃 gives sin 𝜃 is equal to 150 divided by 3168. To calculate the value of 𝜃, we work out inverse sin of 180 divided by 3168, which is 2.713. The angle of inclination of the road is 2.71 degrees correct to two decimal places.