Problem 87 - Precalculus : Building Concepts and Connections

Question

In this set of exercises, you will use degree and radian measure to study real-world problems. In the 1800 s, women often carried pleated fans. One of the fans on display at the Smithsonian is 7 inches long and, when fully open, sweeps out an angle of \(80^{\circ}\) How long is the trim, to the nearest tenth of an inch, on the curved edge of the fan?

Step-by-Step Solution

Answer

The length of the trim on the curved edge of the fan is approximately 9.8 inches.

Step 1: Convert angle from degrees to radians

The angle swept by the fan which is \(80^{\circ}\) needs to be converted into radians by multiplying it with \(\pi/180\): \n\[\(80^{\circ}\) * \(\pi/180\) = \(1.39626\) radians\]

Step 2: Calculate the arc length

Next, the formula \(s = r\theta\) for arc length is applied to find out the length of the trim on the curved edge of the fan. Here, \(r = 7\) inches (length of the fan) and \(\theta = 1.39626\) radians. Substituting these values in formula will give: \n\[s = 7 * 1.39626 = 9.77382\] inches

Step 3: Round off to nearest tenth

The length of the fan's trim is approximately 9.77382 inches. Need to round off the value to the nearest tenth which gives: \n\[9.8 inches\]

Key Concepts

Degree Measurement

When studying angles in trigonometry, the degree measurement is a crucial way for quantifying the size of an angle. A degree, indicated by the symbol \(^{\circ}\), fractionates a circle into 360 equal parts. This means that a full circle is 360 degrees. In practical terms, this makes it easier to visualize and measure angles in various contexts.
A degree is useful when evaluating scenarios such as navigation and engineering, as it allows for straightforward computations and interpretations. Converting degree measurement to other units can help perform diverse operations, such as radian conversion, which we'll discuss in the next section. Grasping the concept of degrees enables the solving of complex trigonometric and real-life problems, like determining the trim length on a fan.

Radian Conversion

In trigonometry, converting degree measurements to radians is essential for performing calculations, especially when dealing with circular objects. Radian is the standard unit of angular measure in mathematics and is widely used since it simplifies many formulas.

• 1 radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
• The total radians in a full circle is \(2\pi\).

To convert degrees to radians, the formula used is: \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\).
For example, converting \(80^{\circ}\) as done in the exercise involves multiplying by \(\frac{\pi}{180}\), resulting in approximately \(1.39626\) radians. Understanding this conversion is indispensable for arc length calculations.

Arc Length Calculation

Once the angle is known in radians and the radius of the circle is provided, calculating the arc length becomes straightforward. The arc length formula \(s = r\theta\) links the radius \(r\) and the angle \(\theta\) given in radians to determine the arc length \(s\).

• \(r\) represents the radius of the circle or arc, which in the case of the fan problem is 7 inches.
• \(\theta\) is the angle of the arc expressed in radians, \(1.39626\) radians for this exercise.
• \(s\) denotes the length of the trim along the curved edge.

By substituting these values into the formula, getting \(s = 7 \times 1.39626\) yields \(9.77382\) inches.
Finally, rounding to the nearest tenth gives \(9.8\) inches. This calculation provides a tangible application of trigonometry concepts to everyday objects, enhancing your understanding of how mathematics relates to real-world examples.