Problem 68
Question
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 -ft slope inclined at \(30^{\circ}\).
Step-by-Step Solution
Verified Answer
It takes approximately 15.81 seconds to slide down the slope.
1Step 1: Identify Given Values
We are given that the length of the slope, \(d\), is 2000 feet and the angle of inclination, \(\theta\), is \(30^{\circ}\).
2Step 2: Substitute Values into Formula
Substitute the given values into the time formula: \[ t = \sqrt{\frac{2000}{16 \sin(30^{\circ})}} \] where \(d = 2000\) feet and \(\theta = 30^{\circ}\).
3Step 3: Calculate \( \sin(30^{\circ}) \)
The sine of \(30^{\circ}\) is a known trigonometric value: \( \sin(30^{\circ}) = \frac{1}{2} \).
4Step 4: Simplify and Calculate the Denominator
Substitute \( \sin(30^{\circ}) = \frac{1}{2} \) into the equation:\[ 16 \times \frac{1}{2} = 8 \] Now the formula becomes:\[ t = \sqrt{\frac{2000}{8}} \]
5Step 5: Simplify the Expression Inside the Square Root
Calculate \(\frac{2000}{8}\): \( \frac{2000}{8} = 250 \). This simplifies the formula to:\[ t = \sqrt{250} \]
6Step 6: Compute the Square Root
Calculate \( \sqrt{250} \), which approximately equals 15.81 seconds.
Key Concepts
Angle of InclinationSine FunctionSlope CalculationTime Calculation
Angle of Inclination
The angle of inclination refers to the angle between a horizontal surface and another surface, such as a slope or a road. In trigonometry, this angle is essential when evaluating how a slope behaves because it helps in understanding its steepness.
For example, in our problem, the angle of inclination is given as \(30^{\circ}\). This means the slope makes a 30-degree angle with the horizontal ground.
When you're dealing with slopes, a larger angle of inclination indicates a steeper slope, which affects how objects will move along this surface. Small angles tell us that the slope is gentler. Understanding this concept is crucial in various fields like physics and engineering, where the behavior of sliding objects over different inclinations is observed.
For example, in our problem, the angle of inclination is given as \(30^{\circ}\). This means the slope makes a 30-degree angle with the horizontal ground.
When you're dealing with slopes, a larger angle of inclination indicates a steeper slope, which affects how objects will move along this surface. Small angles tell us that the slope is gentler. Understanding this concept is crucial in various fields like physics and engineering, where the behavior of sliding objects over different inclinations is observed.
Sine Function
The sine function is a fundamental concept in trigonometry that relates to angles within a right triangle. For any angle \(\theta\), it gives the ratio of the length of the side opposite the angle to the hypotenuse.
In our exercise, we make use of \( \sin(30^{\circ}) \). The sine of 30° is a commonly known value: \(\sin(30^{\circ}) = \frac{1}{2}\).
Knowing the sine values of standard angles (like 30°, 45°, 60°) is quite useful, as these values frequently pop up in trigonometry problems. The sine function is integral in calculating aspects of triangles, wave properties, and even simple harmonic motion.
In our exercise, we make use of \( \sin(30^{\circ}) \). The sine of 30° is a commonly known value: \(\sin(30^{\circ}) = \frac{1}{2}\).
Knowing the sine values of standard angles (like 30°, 45°, 60°) is quite useful, as these values frequently pop up in trigonometry problems. The sine function is integral in calculating aspects of triangles, wave properties, and even simple harmonic motion.
Slope Calculation
Calculating the slope is crucial for determining how steep a surface is. In our problem, the slope is not just measured in terms of rise/run but also by using trigonometric functions like sine in the context of a given angle.
The formula we used, \(t = \sqrt{\frac{d}{16 \sin \theta}}\), incorporates the sine function to account for how the angle of the slope impacts the slide time.
To find how steep our hillside is at 30°, we solve \(16 \times \sin(30^{\circ})\) to simplify the calculation process. This tells us that slopes not only vary by elevation difference but also are greatly influenced by the angle of inclination, which can be calculated through trigonometry.
The formula we used, \(t = \sqrt{\frac{d}{16 \sin \theta}}\), incorporates the sine function to account for how the angle of the slope impacts the slide time.
To find how steep our hillside is at 30°, we solve \(16 \times \sin(30^{\circ})\) to simplify the calculation process. This tells us that slopes not only vary by elevation difference but also are greatly influenced by the angle of inclination, which can be calculated through trigonometry.
Time Calculation
Time calculation in this exercise revolves around determining how long it takes for a sled to slide down a slope. We use the formula \(t = \sqrt{\frac{d}{16 \sin \theta}}\) where \(d\) is the slope distance and \(\theta\) is the angle.
Inserting the given values for a 2000-foot slope inclined at \(30^{\circ}\), the first step is to calculate \(\sin(30^{\circ})\). Knowing that \(\sin(30^{\circ}) = \frac{1}{2}\), we compute the expression under the square root and simplify it: \(\frac{2000}{8} = 250\).
Finally, finding \(\sqrt{250}\) gives us approximately \(15.81\) seconds. Thus, the time it takes, in practical terms, is how fast the sled can slide down given the slope's length and inclination angle.
Inserting the given values for a 2000-foot slope inclined at \(30^{\circ}\), the first step is to calculate \(\sin(30^{\circ})\). Knowing that \(\sin(30^{\circ}) = \frac{1}{2}\), we compute the expression under the square root and simplify it: \(\frac{2000}{8} = 250\).
Finally, finding \(\sqrt{250}\) gives us approximately \(15.81\) seconds. Thus, the time it takes, in practical terms, is how fast the sled can slide down given the slope's length and inclination angle.
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