Question

I need to know the answer... please help

Answer 1

The diagram shows that the area we are trying to find is the area of the bigger circle - the area of the smaller circle. If we can find the areas of both of these shapes, then we can find the area of the darkened region.

First, let's try to find the area of the smaller region. Remember that the formula for the area of a circle is $A = \pi r^2$. The radius of the smaller circle is (2x - 3). Thus, the area of the smaller circle is ($A_s$ is just the notation for the area of the smaller circle):

$A_s = \pi (2x - 3)^2$

$A_s = \pi (4x^2 - 12x + 9)$

Now, the area of the larger circle can be found with the same formula. The radius is (2x + 3). Thus, the area of the larger circle is:

$A_l = \pi (2x + 3)^2$

$A_l = \pi (4x^2 + 12x + 9)$

Now, let's subtract the area of the smaller circle from the area of the larger circle.

$A_l - A_s$

$A_l - A_s = [\pi (4x^2 + 12x+ 9)] - [\pi (4x^2 - 12x + 9)]$

$A_l - A_s = \pi [(4x^2 + 12x + 9) - (4x^2 - 12x + 9)]$

$A_l - A_s = \pi [24x]$

$\boxed{A_l - A_s = 24 \pi x}$

The area of the shaded region is 24πx.