Problem 78 - Algebra and Trigonometry

Question

A road is inclined at an angle of 5°. After driving 5000 feet along this road, find the driver’s increase in altitude. Round to the nearest foot.

Step-by-Step Solution

Answer

The increase in altitude is approximately 436 feet to the nearest foot.

Step 1: Identify Important Information

It is given that the angle of inclination of the road is 5° and the distance driven along the road, which acts as the hypotenuse, is 5000 feet.

Step 2: Set up the Equation

In a right-angle triangle, we know that sine(theta) = opposite/hypotenuse. A similar relationship can be built for this problem, with opposite representing the altitude increase (to be solved for), and hypotenuse representing the driven road distance. So, sine(5°) = Altitude/5000 feet.

Step 3: Solve for Altitude

Multiplying both sides by 5000 feet, the equation now reads: Altitude = 5000 feet * sine(5°). Calculating for sine(5°) gives approximately 0.0872, and multiplying this by 5000 feet gives the approximate altitude.

Step 4: Round to the Nearest Foot

Round the obtained altitude to the nearest foot as the problem requires.

Key Concepts

Angle of Inclination

The angle of inclination refers to the angle formed by a surface or line relative to the horizontal plane. In our problem, the road is inclined at an angle of 5 degrees. This means that the road gently slopes upward. The angle of inclination helps us understand how steep or gentle a road is.

It can be visualized as the angle at which two lines meet, where one line runs along the slope of the road, and the other is a level horizontal. This concept is crucial when applying trigonometry to find distances or altitudes. By knowing the angle, we can calculate other measurements, like how high above the ground one travels after moving a given distance along the slope.

Right-Angle Triangle

A right-angle triangle is a triangle in which one of the angles measures exactly 90 degrees. This is the foundation of many trigonometric calculations.

In the context of our problem:

• The hypotenuse is the road you drive along, measuring 5000 feet in length.
• The opposite side of the triangle, also known as the rise or increase in altitude, is what we aim to find.
• The angle of inclination provides the angle at the base of this triangle.

Understanding how these components relate within a right-angle triangle simplifies solving many geometric problems, particularly when calculating angles and side lengths.

Sine Function

The sine function is a crucial part of trigonometry, linking the angle of a triangle to the ratio of the lengths of its sides. Specifically, for a given angle in a right-angle triangle, the sine function is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

In mathematical terms, it is expressed as: \[\text{sine} (\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\]
For our problem, we use \( \theta = 5^{\circ} \), the opposite side being the altitude we seek, and the hypotenuse being the 5000 feet along the road. Thus, we set up the equation as:\[\text{sine} (5^{\circ}) = \frac{\text{Altitude}}{5000}\text{ feet}\]
This equation will help us calculate the exact increase in altitude by isolating the altitude on one side of the formula.

Altitude Calculation

To find the altitude increase, we rearrange our equation from the sine function to solve for the altitude. Substituting \( \text{sine} (5^{\circ}) \approx 0.0872 \), we multiply both sides by the hypotenuse (5000 feet):
\[\text{Altitude} = 5000 \text{ feet} \times 0.0872\]
This calculation gives us the increase in altitude when driving along this inclined road. Once you compute this multiplication, you get approximately 436 feet. Rounding this result to the nearest foot gives us the final altitude increase.
Understanding how to execute this calculation is essential for solving many trigonometry-related problems, as it involves applying concepts of angle measurements with defined formulas.