Problem 68 - 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 \(2000-\mathrm{ft}\) slope inclined at \(30^{\circ} .\)

Step-by-Step Solution

Answer

The sledding time is approximately 15.81 seconds.

Step 1: Understand the Given Problem

Identify the key components of the problem: The slope length \(d\) is 2000 feet and the incline angle \(\theta\) is 30 degrees. We need to compute the time \(t\) it takes for the sled to slide down the slope.

Step 2: Set Up the Formula

The formula provided for calculating the time is: \[ t = \sqrt{\frac{d}{16 \sin \theta}} \]. Substitute \(d = 2000\) feet and \(\theta = 30^\circ\) into this formula.

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

We need to find \(\sin 30^\circ\). We know from trigonometric values that \(\sin 30^\circ = \frac{1}{2}\).

Step 4: Substitute Values into the Formula

Substitute \(d = 2000\) and \(\sin 30^\circ = \frac{1}{2}\) into the formula: \[ t = \sqrt{\frac{2000}{16 \times \frac{1}{2}}} \].

Step 5: Simplify the Expression

Simplify the expression inside the square root: \[ 16 \times \frac{1}{2} = 8 \]. Therefore, we have \[ t = \sqrt{\frac{2000}{8}} \].

Step 6: Compute the Final Answer

Now compute \(\frac{2000}{8} = 250\), and then the square root: \[ t = \sqrt{250} \approx 15.81 \]. Hence, the time taken is approximately 15.81 seconds.

Key Concepts

Inclined Plane

An inclined plane is a flat surface tilted at an angle to the horizontal. It makes transporting an object vertically easier by spreading the work over a distance. The concept is practical in real-life scenarios such as ramps or hills. Inclined planes help reduce the force required to move an object upwards by increasing the distance over which the force is applied.

• They are one of the six classical simple machines.
• The angle of inclination ( \(\theta\)) is crucial in determining the effort needed to move an object.

In the given problem, the inclined plane is described as a hillside. The angle at which it is inclined impacts how fast the sled moves down the slope. A steeper angle will generally result in a faster descent because gravity has a more direct path to act on the sled. This is why understanding inclined planes is key in solving such problems in trigonometry.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent help relate angles to side lengths in a right triangle. In this problem, the function utilized is the sine function, which relates the angle of the plane to the vertical aspect of the slope.

• The sine function for an angle is defined as the opposite side over the hypotenuse in a right triangle.
• We're interested in \(\sin \theta\), which helps account for the gravitational component down the slope.

For \(30^{\circ}\), specifically, \(\sin 30^{\circ} = \frac{1}{2}\). This known trigonometric value simplifies the computation. By substituting the sine value into the formula, we can find how the angle influences the sled's speed down the slope. Knowing trigonometric values for common angles is essential in solving these types of problems efficiently.

Square Roots

Square roots are a mathematical operation that helps us find what number, multiplied by itself, produces the original number. They often appear in formulas dealing with physics and math to relate distances or angles to other measurements.

• The square root symbol is \(\sqrt{}\), and using it involves finding a number which, when multiplied by itself, equals the number inside the symbol.
• In the problem, we calculate \(\sqrt{250}\) to find how long it takes the sled to slide.

Computing the square root of a number might require a calculator or an estimation for non-perfect squares. In our case, \(\sqrt{250}\) is around 15.81. Understanding how to simplify and calculate square roots allows you to find solutions involving distances, times, and rates of motion, making it a crucial skill in trigonometry and physics.