Answer:
The area of the rectangle is [tex]500\:{cm^2}[/tex].
Step-by-step explanation:
It is easier to solve the problem when an illustration is presented. Refer to the illustration at the bottom.
Semicircle is defined as half of the circle. This means that if the area of the circle is defined as [tex]A=\pi r^2[/tex], then the area of a semicircle is [tex]A_{semicircle}=\frac{\pi r^2}{2}[/tex].
1. It is one-dimensional.
2. It's area is half of the circle.
3. Its radius is half of its diameter.
4. It's part of a circle.
The problem said that the circle is inscribed inside the rectangle, this means that the semicircle is contained inside the rectangle. Refer to the illustration.
The area of the rectangle is defined as [tex]A=l\times{w}[/tex]. So to solve the area of the rectangle we need to solve first its length and width. Notice that the diameter of the semicircle serves as the length of the rectangle and the radius of the semicircle serves as the width of the rectangle. This means that we need to find the value of the radius and diameter for us to solve for the area of the rectangle. Remember also that the diameter is twice the length of the radius.
1. Substitute the value of [tex]A_{semicircle}=1250\pi\\[/tex] into the semicircle area formula above.
2. Do cross multiplication.
3. Divide both sides by [tex]\pi\\[/tex].
4. Take the square root of both sides.
5. Take the positive value only since distance or length cannot be less than zero.
6. Multiply the radius value by 2 to solve for the diameter.
[tex]\begin{aligned}1250\pi&=\frac{\pi{r^2}}{2}\\2500\pi&=\pi{r^2}\\\frac{2500\pi}{\pi}&=\frac{\pi{r^2}}{\pi}\\2500&=r^2\\\pm\sqrt{2500}&=\sqrt{r^2}\\50&=r\end{aligned}[/tex]
This means that the radius of the semicircle is 50 cm (width of the rectangle) and the diameter of the semicircle is 100 cm(length of the rectangle).
Now, we can solve for the area of the rectangle. Substitute the values [tex]l=100[/tex] and [tex]w=50[/tex] into the rectangle area formula, then simplify.
[tex]\begin{aligned}A&=100\times{5}\\&=500\:{cm^2}\end{aligned}[/tex]
In conclusion, the area of the rectangle is [tex]500\:{cm^2}[/tex]
To see other examples that deal with rectangular area, go to https://brainly.ph/question/110308
To see the area of the circle, access https://brainly.ph/question/105449
To know more details about semicircle, click https://brainly.ph/question/942914
