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.
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To express the area enclosed by the track as a function of its semicircular radius, we need to find the areas of the two semicircles and the rectangle in between them.
Let's assume the radius of the semicircle is represented by r.
Area of a circle = π * r^2
Area of a semicircle = (1/2) * area of a circle = (1/2) * π * r^2
Area of one semicircle = (1/2) * π * r^2
Since there are two semicircles, the total area of the two semicircles is:
2 * (1/2) * π * r^2 = π * r^2
The area of the rectangle can be calculated by multiplying the length of the straight lines between the two semicircles by the width, which is equal to the diameter (2 * r):
Area of the rectangle = length * width = 2r * 2r = 4r^2
Therefore, the total area enclosed by the track is the sum of the areas of the two semicircles and the rectangle:
Total area = π * r^2 + 4r^2
Simplifying the expression:
Total area = π * r^2 + 4r^2
Total area = (π + 4) * r^2
The domain of the function is the set of all possible values for the radius, r. In this case, since we are dealing with a racetrack, the radius cannot be negative, so the domain is the set of all nonnegative real numbers:
Domain = [0, ∞)
To express the area enclosed by the racetrack as a function of its semicircular radius, we need to break down the shape into its individual components and then find the total area.
First, let's consider that there are two semicircular ends connected by two straight lines. Each semicircle will have a radius, which we can denote as "r." The straight lines connecting the semicircles will have a length of 2r (since they connect two points on the circumference of each semicircle).
Now, let's find the area of each component:
1) Area of one semicircle:
The formula for the area of a semicircle is (π * r^2) / 2. Therefore, the area of one semicircle is (π * r^2) / 2.
2) Area of the two straight lines:
The length of each straight line is 2r, so the total length of both straight lines is 4r. Since the width of each straight line is negligible, we can multiply the total length by the width of the track to get the area. Let's assume the width of the track is denoted by "w." Therefore, the area of the two straight lines is 4r * w.
Now, let's add up the areas of the two semicircles and the two straight lines to find the total area of the track:
Total Area = 2 * (Area of one semicircle) + Area of the two straight lines
= 2 * ((π * r^2) / 2) + (4r * w)
= π * r^2 + 4r * w
So, the area enclosed by the racetrack is given by the function A(r) = π * r^2 + 4r * w, where "r" is the radius of the semicircle and "w" is the width of the straight lines connecting the semicircles.
Now, let's determine the domain of the function A(r). In this context, the radius "r" cannot be negative, so the domain of the function A(r) is r ≥ 0.