a 1 mile racetrack has two semicircula ends connected by straight lines.express the area enclosed by the track as a function of its semi circular radius.determine its domain.
To find the area enclosed by the racetrack, we need to calculate the areas of the two semi-circular ends and the rectangular middle section.
1. Area of the semi-circular ends:
The formula for the area of a circle is A = πr^2, where A is the area and r is the radius. Since we have semi-circles, we need to divide the formula by 2. So the area of one semi-circular end is A_end = (πr^2) / 2.
2. Area of the rectangular middle section:
The length of the rectangular section is the entire track length minus the circumference of the two semi-circular ends. Since the track length is 1 mile, the length of the middle section is L = 1 - 2πr.
The width of the rectangular middle section is the diameter of the semi-circular ends, which is 2r.
Therefore, the area of the rectangular middle section is A_mid = L * width = (1 - 2πr) * 2r = 2r - 4πr^2.
3. Total area of the track:
The total area is the sum of the areas of the two semi-circular ends and the rectangular middle section. So, A_total = 2 * A_end + A_mid.
Substituting the respective formulas, we get A_total = πr^2 + 2r - 4πr^2.
The domain of the function, which represents the range of possible values for the radius, depends on the physical constraints of the racetrack. Commonly, the radius cannot be negative, so the domain is all non-negative real numbers: r ≥ 0.