Question

Designing a Little League Field For a 60 -foot Little League Baseball field, the distance from home base to the nearest fence (or other obstruction) in fair territory should be a minimum of 200 feet. The commissioner of parks and recreation is making plans for a new 60 -foot field. Because of limited ground availability, he will use the minimum required distance to the outficld fence. To increase safety, however, he plans to include a 10 -foot wide warning track on the inside of the fence. To further increase safety, the fence and warning track will extend both directions into foul territory. In total, the arc formed by the outfield fence (including the extensions into the foul territories) will be subtended by a central angle at home plate measuring \(96^{\circ}\), as illustrated. (a) Determine the length of the outfield fence. (b) Determine the area of the warning track.

Short answer

Arc length of the outfield fence is approx. 335.1 feet. Area of the warning track is approx. 3266.0 square feet.

Step-by-step solution

Understanding the Geometry

The outfield fence is part of a circular arc with a radius of 200 feet extending into foul territories, forming an arc subtended by a central angle of 96 degrees at home plate.

Converting Degrees to Radians

For calculations involving arc length and area, convert the central angle from degrees to radians: \[θ = 96^{\text{o}} \times \frac{\pi}{180} = \frac{8\pi}{15} \]

Calculating the Arc Length for the Fence

Use the formula for arc length, which is given by \( s = rθ \): \[ s = 200 \times \frac{8\pi}{15} = \frac{1600\pi}{15} \approx 335.1 \text{ feet} \]

Determining the Radius of the Warning Track

Since the warning track is 10 feet wide, the radius at the inner edge of the warning track is 190 feet, and the radius at the outer edge is 200 feet.

Calculating the Area of the Warning Track

The area of the warning track is the difference in areas of the two sectors formed by the two radii: \[ A = \frac{1}{2} θ (R_o^2 - R_i^2) \] where: \( R_o = 200 \) and \( R_i = 190 \). Substitute the values: \[ A = \frac{1}{2} \times \frac{8\pi}{15} \times (200^2 - 190^2) = \frac{4\pi}{15} \times (40000 - 36100) = \frac{4\pi}{15} \times 3900 = 1040 \pi \approx 3266.0 \text{ square feet} \]

Key concepts

arc length

The arc length represents the distance along the curved part of a circle or, in this context, the outfield fence of a Little League field. Imagine part of a circle's edge; this is the arc. Here, calculating the arc length involves understanding how much of the circular boundary the outfield fence occupies given a central angle of 96 degrees. We use radians for ease of calculations. One full circle is divided into 360 degrees, but in radians, this full circle is represented as \(2\pi\).

• Step 1: Converting the central angle from degrees to radians is crucial. We use the conversion formula: \(\theta = \frac{\pi}{180} \times \text{degrees}\).
• Step 2: Once we have the angle in radians, we can use the arc length formula \(s = r\theta\), where \(r\) is the radius (200 feet in this case) and \(\theta\) is the angle in radians.
• Step 3: Multiply the radius by the central angle in radians to get the length of the arc. Here, the calculation was: \( s = 200 \times \frac{8\pi}{15} = 335.1 \) feet approximately.

Remember, understanding these steps and concepts will not only help in solving this problem but also similar ones involving circular and angular measurements.

trigonometric angles

Trigonometric angles play a vital role in calculating lengths and areas in sectors of circles. The main idea is to understand how to convert degrees into radians and use these angles for further calculations.

• Step 1: Converting degrees to radians: most of the time, angles are provided in degrees, but for calculations, radians are more effective since they directly relate to the radius and arc length of a circle. The conversion formula \(\theta = \frac{\pi}{180} \times \text{degrees}\) is used often. In this solution, the 96-degree angle was converted as \(96^{\circ} \times \frac{\pi}{180} = \frac{8\pi}{15}\).
• Step 2: Using radians in formulae: in problems involving circles, radians allow us to more easily calculate arc lengths and areas, simplifying the use of formulas such as \(s = r\theta\) for arc length and \(A = \frac{1}{2} \theta (R_o^2 - R_i^2)\) for the area of sectors.

Knowing how to handle trigonometric angles is a foundational skill in geometry and many practical applications, such as designing sports fields or even in navigation.

area of sectors

The area of sectors principle helps us determine the space within a segment of a circle, like the warning track on the Little League field. Sectors are like slices of a pie, where the pie's radius reaches out to form an angle at the center. Here’s how we approach it:

• Step 1: Understanding the radii: the warning track extends from 190 feet to 200 feet in radius. This means you need to calculate the area for both radii and find the difference.
• Step 2: Using the formula for the area of sectors: The area for one sector is given by \(A = \frac{1}{2} \theta r^2\). We need to apply this formula twice, once for the outer radius (200 feet) and once for the inner radius (190 feet).
• Step 3: Subtracting the smaller sector area from the larger: As calculated in the solution, we find \( A = \frac{1}{2} \times \frac{8\pi}{15} \times (200^2 - 190^2) \). This involves finding the difference between the squares of the radii, then applying the sector area formula.

By completing these steps, you get the area of the warning track to be approximately 3266.0 square feet. Thus, mastering the area of sectors is incredibly useful when dealing with any circular designs or applications.