Problem 60 - Algebra 2 and Trigonometry

Question

The bed of a truck is 4.2 feet above the ground. In order to unload boxes from the truck, the driver places a board that is 12 feet long from the bed of the truck to the ground. Find, to the nearest minute, the measure the board makes with the ground.

Step-by-Step Solution

Answer

The board forms an angle of approximately 20 degrees and 20 minutes with the ground.

Step 1: Understand the Problem

We need to find the angle that a ramp (the board) makes with the ground. The truck bed is 4.2 feet above the ground, and the board is 12 feet long.

Step 2: Identify the Triangle

The truck bed, board, and ground form a right triangle where the ramp is the hypotenuse (12 feet), the height is the opposite side (4.2 feet), and the ground is the adjacent side.

Step 3: Use Trigonometry to Find the Angle

Use the sine function to find the angle \( \theta \) that the board makes with the ground. We use \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4.2}{12} \).

Step 4: Calculate \( \theta \)

Calculate \( \theta \) by finding the inverse sine: \( \theta = \sin^{-1}\left(\frac{4.2}{12}\right) \). Use a calculator to find \( \theta \approx 20.33 \) degrees.

Step 5: Convert Degrees to Minutes

To convert degrees to minutes, note that one degree equals 60 minutes. Multiply the decimal portion by 60: 0.33 degrees \( \times 60 = 19.8 \) minutes.

Key Concepts

Sine Function

The sine function is one of the fundamental components of trigonometry, especially when dealing with right triangles. Imagine a right triangle consisting of an angle, which we'll call \( \theta \). The sine of \( \theta \) is calculated as the ratio of the length of the

• Side opposite the angle \( \theta \)
• Hypotenuse, the triangle's longest side

If you're tackling a problem involving the sine function, writing it simply as \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \) is your go-to formula. For example, in the truck bed problem, the height from the ground to the bed forms the opposite side, while the board is the hypotenuse. When you set this into the formula, it provides an easy and effective route to identify unknown angles. This is key in many real-world problems and is a powerful tool in both small calculations and more complex geometrical reasoning.

Inverse Sine

When you need an angle from the sides of a triangle, inverse trigonometric functions become handy. The inverse sine, also expressed as \( \sin^{-1} \) or \( \text{arcsin} \), allows you to find an angle when you know the ratio of the opposite side to the hypotenuse. For instance, in our previous example of the truck, once you calculate \( \sin(\theta) = \frac{4.2}{12} \), determine \( \theta \) by computing \( \theta = \sin^{-1}\left(\frac{4.2}{12}\right) \). This gives you the angle \( \theta \) in degrees. Calculators often have this function, and they are vital for seamlessly converting known ratios into useful angle measures. They help directly connect the mathematical world to practical applications like finding the slope of the board in this scenario.

Degree to Minute Conversion

Degrees are subdivided into smaller parts, like minutes, which facilitate more precise measurements of angles. Every degree contains 60 minutes. To convert a fractional degree into minutes, simply multiply the fraction by 60.In the truck problem, if \( \theta \approx 20.33 \) degrees, isolate the decimal part, \( 0.33 \), and proceed to convert it. Multiply \( 0.33 \) by 60 to get \( 19.8 \) minutes. The outcome is

• 20 degrees
• 19.8 minutes

Rounding to the nearest whole number gives approximately 20 degrees and 20 minutes. This precision level is useful in scenarios that need accuracy beyond whole degrees.