Question

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

Answer 1

Answer: $2x^2+\frac{\pi x^2}{4}$

Step-by-step explanation:

The area of a rectangle can be calculated with this formula:

$A_r=lw$

Where "l" is the lenght and "w" is the width.

The area of a semi-circle can be found with this formula:

$A_{sc}=\frac{\pi r^2}{2}$

Where "r" is the radius.

 Let be "x" the shorter side of the rectangle (which is also the diameter of the semi-circle).

Based on the information given in the exercise, you know that:

$w=x\\\\l=2x$

Since the radius is half the diameter:

$r=\frac{x}{2}$

Observe the figure attached, which shows the stadium.

The area of the stadium is the sum of the areas of the semi-cirlcles and the rectangle.

Therefore, you can construct the following expression for the area of the stadium is terms of "x":

$(2x)(x)+\frac{\pi (\frac{x}{2})^2}{2}+\frac{\pi (\frac{x}{2})^2}{2}$

Simplifying it, you get:

$(2x)(x)+\frac{\frac{\pi x^2}{4}}{2}+\frac{\frac{\pi x^2}{4}}{2}=2x^2+\frac{\pi x^2}{4}$