Problem 65 - Precalculus with Limits

Question

ROAD GRADE A straight road rises with an inclination of 0.10 radian from the horizontal (see figure). Find the slope of the road and the change in elevation over a two-mile stretch of the road.

Step-by-Step Solution

Answer

The slope of the road is approximately \(0.10\) and the change in elevation over a two-mile stretch of the road is approximately \(0.20\) miles.

Step 1: Find the Slope of the Road

To find the slope, calculate the tangent of the angle of inclination, which in this case is \(0.10\) radian. The slope of the road is given by \(\tan(0.10)\). Calculate this value to obtain the slope.

Step 2: Find the Change in Elevation

To find the change in elevation over a two-mile stretch of road, use the sine of the angle of inclination. The sine gives us the ratio of the change in elevation (height) to the length of the road. Therefore, the change in elevation is given by \(\sin(0.10) * 2\) miles. Calculate this value to get the change in elevation over a two-mile stretch.

Key Concepts

Slope of a Road

Understanding the slope of a road is important in both practical and mathematical contexts. When you think about the slope of a road, you're considering how steep the road is. In trigonometry, the slope is linked to the tangent of the angle of inclination.
To calculate the slope of a road, you use the formula:

• \( \text{slope} = \tan(\text{angle of inclination}) \).

The tangent function relates the angle of the road's rise compared to a flat horizontal surface.
For example, if the angle of inclination is given as 0.10 radians, you find the slope by calculating \( \tan(0.10) \). This value gives you a measure of how much the road rises in comparison to its horizontal distance, helping to determine how steep the road is.

Angle of Inclination

The angle of inclination is a measure of how steep a road is compared to a perfectly flat, horizontal line. It is an essential concept for engineers and construction workers who design roads.
Trigonometry often uses this angle to further calculate slope and elevation changes. You might think of it as the tilt of the road. In our example, a road has an angle of inclination of 0.10 radians.
Radian is a unit used in measuring angles, with one radian equivalent to approximately 57.3 degrees.

• Knowing the angle of inclination helps determine the slope.
• It also helps in figuring out the change in elevation over a given distance.

With this angle, you can use trigonometric functions like sine and tangent to calculate various aspects of the road, such as its steepness and vertical rise.

Change in Elevation

The change in elevation describes how much the road increases vertically over a certain horizontal distance. This is particularly useful for determining how high you will go when traveling on an inclined road.
To find the change in elevation, you use the sine function of the angle of inclination:

• \( \text{change in elevation} = \sin(\text{angle of inclination}) \times \text{horizontal distance} \).

For a two-mile road with a 0.10 radian inclination, we calculate this by \( \sin(0.10) \times 2 \text{ miles} \).
The sine function helps us understand the "opposite" side of the triangle formed by the road's slope, allowing us to compute how much vertical distance is covered.
Therefore, the change in elevation is not just about how far you travel horizontally but also how high you get along the way.