Find the approximate area of the shaded region. The two circles are congruent to each other and tangent to the rectangle and each other.

Full solution

Q. Find the approximate area of the shaded region. The two circles are congruent to each other and tangent to the rectangle and each other.

Identify Dimensions: Identify the dimensions of the rectangle. Since the circles are congruent and tangent to the rectangle, the length of the rectangle is equal to the diameter of one circle, and the width is equal to the radius.
Calculate Radius: Calculate the radius of the circles. $\newline$ Assume the radius of each circle is $r$ . The length of the rectangle is $2r$ , and the width is $r$ .
Calculate Rectangle Area: Calculate the area of the rectangle. $\newline$ Area of rectangle = $\text{length} \times \text{width} = 2r \times r = 2r^2$ .
Calculate Circle Area: Calculate the area of one circle. $\newline$ Area of circle = $\pi \times r^2$ .
Calculate Total Circle Area: Calculate the total area of both circles. $\newline$ Total area of circles = $2 \times (\pi \times r^2) = 2\pi r^2$ .
Calculate Shaded Area: Calculate the area of the shaded region. $\newline$ Area of shaded region = Area of rectangle - Total area of circles = $2r^2 - 2\pi r^2$ .
Substitute and Find Area: Substitute the value of $r$ to find the area. $\newline$ Let's say $r = 5$ (for example). Then the area of the shaded region would be $2(5)^2 - 2\pi(5)^2 = 50 - 50\pi$ .