Find the area of the shaded region (everything but the circles). Express your answer in terms of . Use pi to represent the symbol For example: would be typed as 32pi

The area of the shaded region is (540 - 65.25pi) in²

Step-by-step explanation:

The remaining part of the question which is an image is attached below.

Step-by-step explanation:

To determine the area of the shaded region, that will be the difference between the area of the rectangle and the sum of the areas of the three circles.

First, we will determine the area of the rectangle, which is given by

Area of rectangle = Length × Breadth

In the figure, Breadth = 18 in

and Length = 12 in + 9 in + 6 in + 3 in

∴ Length = 30 in

Hence,

Area of rectangle = 30 in × 18 in

Area of rectangle = 540 in²

Now, we will determine the areas of the circles one after the other

For the smallest circle

From the figure, the diameter of the smallest circle is 6 in

From Radius = Diameter / 2

Hence,

Radius of the smallest circle = 6 in / 2 = 3 in

Area of a circle is given by

Area of circle = πr²

∴ For the smallest circle

Area of the smallest circle = π×3² = 9π in²

Hence, area of the smallest circle is 9pi in²

For the circle in the middle

Diameter of the circle in the middle is 9 in

∴ Radius of the circle in the middle = 9 in / 2 = 4.5 in

For the area of the circle in the middle

Area of the circle in the middle = π × 4.5² = 20.25π in²

Hence, Area of the circle in the middle is 20.25pi in²

For the largest circle

From the figure, the diameter of the largest circle is 12 in

Hence, radius of the largest circle = 12 in / 2 = 6 in

∴ For the largest circle

Area of the largest circle = π×6² = 36π in²

Hence, area of the largest circle is 36pi in²

Sum of the areas of the three circles = 9pi in² + 20.25pi in² + 36pi in²

Sum of the areas of the three circles = 65.25pi in²

Now,

Area of the shaded region = Area of the rectangle - Sum of the areas of the three circles

∴ Area of the shaded region = 540 in² -  65.25pi in²

Area of the shaded region = (540 - 65.25pi) in²

Hence, the area of the shaded region is (540 - 65.25pi) in²