What is the exact area of a track surrounding an athletic field measuring 120 yards by 40 yards with semicircles at both ends, expressed in square yards in terms of π?

To calculate the area of the track surrounding the athletic field, we first need to determine the layout of the field and the attached track. The field measures 120 yards in length and 40 yards in width, and it has semicircles at both ends, making the overall shape of the field with track somewhat like a rectangle with circles at the ends.

1. Calculate the area of the field:

The area of the rectangular athletic field can be calculated using the formula for the area of a rectangle, which is length multiplied by width:

[

\text{Area of field} = 120 \times 40 = 4800 \text{ square yards}

]

2. Determine the dimensions of the surrounding track:

The track has a uniform width. Let's denote the width of the track as ( w ). The dimensions of the entire area, including the track, will then be:

• Length: ( 120 + 2w ) (adding ( w ) to each end)

• Width: ( 40 + 2w ) (adding ( w ) to each side)

3. Calculate the area of the entire track and field:

The