Problem 66 - Algebra and Trigonometry

Question

Sledding The time in seconds that it takes for a sled to slide down a hillside inclined at an angle \(\theta\) is $$t=\sqrt{\frac{d}{16 \sin \theta}}$$ where \(d\) is the length of the slope in feet. Find the time it takes to slide down a \(2000-f t\) slope inclined at \(30^{\circ} .\)

Step-by-Step Solution

Answer

About 15.81 seconds.

Step 1: Understanding the Formula

The formula given is \( t = \sqrt{\frac{d}{16 \sin \theta}} \). This equation calculates the time \( t \) in seconds it takes for a sled to slide down a hillside. The variables are \( d \), which represents the slope's length in feet, and \( \theta \), the angle of inclination in degrees.

Step 2: Identifying Given Values

We know that the length of the slope \( d \) is 2000 feet, and the angle \( \theta \) is \( 30^{\circ} \). We'll use these values to find the time \( t \).

Step 3: Calculating \( \sin \theta \)

For an angle \( \theta = 30^{\circ} \), we use trigonometric tables or a calculator to find \( \sin 30^{\circ} = 0.5 \).

Step 4: Substitute and Simplify

Substitute \( d = 2000 \) and \( \sin 30^{\circ} = 0.5 \) into the formula:\[t = \sqrt{\frac{2000}{16 \times 0.5}} = \sqrt{\frac{2000}{8}}\]Simplifying the fraction, we have:\[\frac{2000}{8} = 250\]

Step 5: Final Calculation

Compute the square root of 250:\[t = \sqrt{250} \approx 15.81\]Thus, the time it takes to slide down the slope is approximately 15.81 seconds.

Key Concepts

Trigonometric functions

Trigonometric functions are essential in understanding relationships within triangles. They help us make sense of angles and distances. In our exercise, we specifically focus on the sine function, denoted as \( \sin \theta \). This function helps relate the angle of inclination with the motion down the slope. By using \( \sin \theta \), we can determine how steep the slope is, which is crucial to finding out how long the sled takes to travel down the hillside.

• **Key Functions**: sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)).
• **Angle and Ratio**: For \( \theta = 30^{\circ} \), \( \sin 30^{\circ} = 0.5 \).

Trigonometric functions turn angles into ratios. These ratios are essential for various calculations in both mathematics and physics, especially when dealing with inclined surfaces.

Inclined plane

An inclined plane is a flat surface tilted at an angle to the horizontal. It is a classical physics problem that helps us explore forces and motion. This incline makes tasks like sliding a sled down a hill more straightforward to analyze. In our scenario, the sled on the inclined plane faces gravity, which helps pull it down the slope.

• **Concept of Inclination**: Helps determine how much of the gravitational force contributes to the sled's motion downwards.
• **Invention of Inclined Plane**: Simplifies calculations of motion by focusing on parallel and perpendicular forces.

Understanding the inclined plane means understanding how various forces, such as gravity, friction, and normal forces, interact when an object is on this surface. The angle \( \theta = 30^{\circ} \) plays a pivotal role in these calculations.

Problem-solving

Problem-solving in trigonometry requires identifying the known values, using the right formula, and simplifying step-by-step. Our sled problem showcases this beautifully:

• **Identify Variables**: Recognize that \( d = 2000 \) feet and \( \theta = 30^{\circ} \).
• **Utilize Formulas**: Use the equation \( t = \sqrt{\frac{d}{16 \sin \theta}} \) to find the desired time.
• **Simplify and Calculate**: Substitute known values and simplify to find \( t \approx 15.81 \) seconds.

Effective problem-solving involves understanding the question, systematically applying relevant mathematical concepts, and working through calculations to find a solution. In trigonometry and physics, breaking problems into smaller steps, as exemplified here, makes even daunting tasks more manageable.