You are tasked with designing a perfectly circular track for testing vehicle performance. The track will be paved and the infield of the track will be covered with astroturf. The requirements are as follows: 1. Track width must be exactly 12.5 meters 2. Inner diameter of track must be between 113 - 186 meters. Design a track that meets these requirements, specifying dimensions. Then determine how many square meters of astroturf and pavement are needed for the project. able
You are tasked with designing a perfectly circular track for testing vehicle
performance. The track will be paved and the infield of the track will be covered with
astroturf. The requirements are as follows:
1. Track width must be exactly 12.5 meters
2. Inner diameter of track must be between 113 - 186 meters.
Design a track that meets these requirements, specifying dimensions. Then
determine how many square meters of astroturf and pavement are needed for the
project.
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Step-by-step explanation:
To design a track that meets the given requirements, we need to determine the outer diameter of the track based on the specified track width.
1. Track width: 12.5 meters
2. Inner diameter: 113 - 186 meters
To calculate the outer diameter, we add twice the track width to the inner diameter:
Outer diameter = Inner diameter + 2 * Track width
Using the minimum inner diameter (113 meters), the outer diameter would be:
Outer diameter = 113 + 2 * 12.5 = 138 meters
Using the maximum inner diameter (186 meters), the outer diameter would be:
Outer diameter = 186 + 2 * 12.5 = 211 meters
Therefore, the dimensions of the track would be a circle with an inner diameter of 113 - 186 meters and an outer diameter of 138 - 211 meters.
To calculate the area of the track, we use the formula for the area of a circular ring:
Area of track = π * (Outer radius^2 - Inner radius^2)
We can calculate the outer and inner radii as follows:
Outer radius = Outer diameter / 2
Inner radius = Inner diameter / 2
Using the minimum and maximum values:
Minimum outer radius = 138 / 2 = 69 meters
Maximum outer radius = 211 / 2 = 105.5 meters
Minimum inner radius = 113 / 2 = 56.5 meters
Maximum inner radius = 186 / 2 = 93 meters
Now we can calculate the area of the track:
Minimum area of track = π * (105.5^2 - 56.5^2)
Maximum area of track = π * (93^2 - 69^2)
To determine the amount of astroturf and pavement needed, we subtract the area of the track from the total area of the outer circle (based on the maximum outer radius):
Total area of outer circle = π * (105.5^2)
Minimum astroturf area = Total area of outer circle - Minimum area of track
Maximum astroturf area = Total area of outer circle - Maximum area of track
The pavement area would be equal to the minimum and maximum area of the track.
Please note that the above calculations assume a perfectly circular track. In practice, adjustments might be required based on the specific terrain and engineering considerations.
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