Chapter 22: Problem 64
Applications A straight highway makes an angle of \(4.50^{\circ}\) with the horizontal. How much does the highway rise in a distance of \(2500 \mathrm{ft}\), measured along the road?
Short Answer
Expert verified
The highway rises approximately 196.66 ft over a distance of 2500 ft.
Step by step solution
01
Convert the angle to radians
To use trigonometrical functions we need to convert the angle from degrees to radians. The conversion formula is: \(\text{radians} = \text{degrees} \times \frac{\pi}{180^\circ} \) So, the angle in radians is \( 4.50^\circ \times \frac{\pi}{180^\circ} \) radians.
02
Use the tangent function
Since the highway makes an angle with the horizontal, we can use the tangent of this angle to find the rise over the run (distance along the highway). The formula is: \( \tan(\theta) = \frac{\text{rise}}{\text{run}} = \frac{\text{rise}}{2500\text{ ft}} \) where \(\theta\) is the angle in radians.
03
Solve for the rise
Rearranging the formula from Step 2 to solve for the rise, we get \( \text{rise} = \tan(\theta) \times 2500\text{ ft} \) Plug in the value of \(\theta\) in radians from Step 1 to calculate the rise.
04
Calculate the rise
Input the value of \(\tan(\theta)\) into the formula and multiply by the run (2500 ft) to find the rise. Perform the calculation using a calculator set to radian mode.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angle Conversion
Before diving into the application of trigonometry in real life, it's crucial to understand angle conversion, especially when working with trigonometric functions. In mathematics, angles can be measured in degrees or radians. While our everyday language and most school problems use degrees, radians are the standard unit of angular measurement in higher mathematics and physics because they simplify the calculations.
For instance, when you're told a highway rises at an angle of \(4.50^\circ\), but you'll be using a trigonometric function that requires the angle in radians, you'll need to convert the degrees to radians using the formula: \[\text{radians} = \text{degrees} \times \frac{\pi}{180^\circ}\]. Simply put, radians provide a more direct measure of angle in terms of the arc length on a unit circle, leading to more intuitive equations and solutions in trigonometry.
For instance, when you're told a highway rises at an angle of \(4.50^\circ\), but you'll be using a trigonometric function that requires the angle in radians, you'll need to convert the degrees to radians using the formula: \[\text{radians} = \text{degrees} \times \frac{\pi}{180^\circ}\]. Simply put, radians provide a more direct measure of angle in terms of the arc length on a unit circle, leading to more intuitive equations and solutions in trigonometry.
Tangent Function
The tangent function is one of the primary trigonometric functions and has a wide range of applications, including solving real-life problems involving angles of elevation or depression. The function relates the angle of a right-angled triangle to the ratio of the opposite side over the adjacent side.
In the context of our highway problem, we are examining a straight road rising at an angle compared to the horizontal plane. The tangent function becomes our tool for relating this angle (after converting it to radians) to the rise of the highway over its run (length along the road). The straightforward relationship given by the function: \(\tan(\theta) = \frac{\text{rise}}{\text{run}}\), enables us to solve for either side as long as we have the angle and one side, establishing this function as indispensable in various fields including engineering and architecture.
In the context of our highway problem, we are examining a straight road rising at an angle compared to the horizontal plane. The tangent function becomes our tool for relating this angle (after converting it to radians) to the rise of the highway over its run (length along the road). The straightforward relationship given by the function: \(\tan(\theta) = \frac{\text{rise}}{\text{run}}\), enables us to solve for either side as long as we have the angle and one side, establishing this function as indispensable in various fields including engineering and architecture.
Calculating Rise Over Run
The concept of 'rise over run' is inherent in understanding slopes, especially when it comes to roads, ramps, or any inclined surfaces. The 'rise' refers to how much an incline gains in height, while the 'run' signifies the horizontal distance over which this gain occurs. It's a practical instance of the tangent function in trigonometry.
To calculate the rise from the given run (distance of the highway), we rearrange our tangent function formula to solve for the rise: \(\text{rise} = \tan(\theta) \times \text{run}\). In our exercise, the run is the known distance of 2500 feet along the highway, and using the angle in radians, we use this formula to determine the height gain, or rise, which is a useful calculation in construction and civil engineering projects.
To calculate the rise from the given run (distance of the highway), we rearrange our tangent function formula to solve for the rise: \(\text{rise} = \tan(\theta) \times \text{run}\). In our exercise, the run is the known distance of 2500 feet along the highway, and using the angle in radians, we use this formula to determine the height gain, or rise, which is a useful calculation in construction and civil engineering projects.
Trigonometry Problems
Solving trigonometry problems often involves a combination of angle conversion, understanding and applying trigonometric functions, and knowing how to translate these calculations into real-world contexts, such as finding the rise of a highway. Recognizing the right function to use and mastering these concepts are key to solving these problems effectively.
For example, when faced with the practical issue of how much a road ascends over a certain distance, we systematically approach this by: converting the angle given in degrees to radians, employing the tangent function to relate this angle to the rise over run, and ultimately calculating the rise that corresponds to our specific problem. This step-by-step method not only simplifies complex trigonometric problems but also translates to a variety of professions such as navigation, surveying, and even graphic design, showcasing the widespread utility of trigonometry in everyday life.
For example, when faced with the practical issue of how much a road ascends over a certain distance, we systematically approach this by: converting the angle given in degrees to radians, employing the tangent function to relate this angle to the rise over run, and ultimately calculating the rise that corresponds to our specific problem. This step-by-step method not only simplifies complex trigonometric problems but also translates to a variety of professions such as navigation, surveying, and even graphic design, showcasing the widespread utility of trigonometry in everyday life.