Question

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

Answer 1

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.