Problem 23 - Contemporary Precalculus

Question

It is claimed that the Ohio Turnpike never has an uphill grade of more than \(3^{\circ} .\) How long must a straight uphill segment of the road be to allow a vertical rise of 450 feet?

Step-by-Step Solution

Answer

Answer: The length of the straight uphill segment of the road is approximately 8610.90 feet.

Step 1: Identify the given information

We are given the following information: 1. The grade of the road: \(3^{\circ}\) 2. The vertical rise: 450 feet

Step 2: Create a right triangle

Draw a right triangle such that the angle A represents the grade of the road, the opposite side a represents the vertical rise (450 feet), and the hypotenuse c represents the length of the uphill segment.

Step 3: Use the sine function

We will use the sine trigonometric function to solve for the length of the hypotenuse (c). Recall the sine function: sin(A) = opposite side (a) / hypotenuse (c) The angle A is given by the grade of the road, \(3^{\circ}\). So, sin(3) = 450/c

Step 4: Solve for the hypotenuse (c)

To find the length of the uphill segment (c), we need to isolate c in the sine function equation: sin(3) = 450/c c = 450/sin(3) Now, calculate the length of the uphill segment by evaluating the expression: c ≈ 450/0.0523 c ≈ 8610.90

Step 5: State the answer

The length of the straight uphill segment of the road must be approximately 8610.90 feet to allow a vertical rise of 450 feet with an uphill grade of no more than \(3^{\circ}\).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, especially right triangles. One of the primary functions in trigonometry is the sine function, which is fundamental for solving problems involving right triangles.

The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse, the longest side across from the right angle. In symbols, for a given angle A, the formula is expressed as \(\text{sin}(A) = \frac{{\text{opposite side}}}{{\text{hypotenuse}}}\). Understanding this function is crucial as it's often used to solve various practical problems involving heights and distances, like the example of calculating the length of a road's uphill segment.

Sine Function Application

The application of the sine function in a real-world context, such as the Ohio Turnpike problem, demonstrates how trigonometry can tackle practical questions. When you know the sine of an angle and the length of one side of a right triangle, you can find an unknown side.

In the Ohio Turnpike problem, we are given a small angle (\(3^{\text{o}}\)) and need to find the length of the road's uphill segment. By applying the sine function, one can easily determine the hypotenuse, representing the road segment, given the opposite side, which represents the rise of the hill. This practical application helps visualize why trigonometry is such an important tool in different fields, including engineering, astronomy, and even navigation.

Right Triangle Problem Solving

Steps for Solving Right Triangle Problems

To solve right triangle problems, there's a series of steps you can follow. First, understand the problem and visualize it with a drawing of a right triangle, labeling known and unknown sides and angles. Next, select the appropriate trigonometric function based on the given and required information.

Then, set up the equation and solve for the unknown value. Finally, interpret the answer and consider if it makes sense in the context of the problem. For the Ohio Turnpike, after calculating the hypotenuse as approximately 8610.90 feet, it's worthwhile to reflect on the solution's reasonableness considering the constraints, such as the maximum uphill grade.