The stadium has a form of a rectangle with two semicircles attached to the short sides. The long side of a rectangle is 2 times longer than the short one.
Construct the expression for the area of the stadium in terms of x, where x denotes a shorter side of the rectangle
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Joe mama
To construct the expression for the area of the stadium, let's break it down into its individual components.
1. The area of the rectangular part: The area of a rectangle is given by the formula A_rect = length × width. We are given that the long side of the rectangle is 2 times longer than the short side. Therefore, the length of the rectangle is 2x and the width is x. Hence, the area of the rectangular part is A_rect = (2x) × x = 2x^2.
2. The area of the two semicircles: The area of a semicircle is given by the formula A_semi = (π × r^2) / 2, where r is the radius of the semicircle. Since the semicircles are attached to the shorter sides of the rectangle, their radius would be equal to half the width of the rectangle, which is x/2. Therefore, the total area of the two semicircles is A_semi_total = 2 × ((π × (x/2)^2) / 2) = 2 × (π × x^2 / 8) = πx^2 / 4.
3. The total area of the stadium: The total area of the stadium is the sum of the area of the rectangular part and the area of the two semicircles. Thus, the expression for the area of the stadium in terms of x is:
A_stadium = A_rect + A_semi_total = 2x^2 + πx^2 / 4.
Therefore, the expression for the area of the stadium in terms of x is 2x^2 + πx^2 / 4.