Q.48 - Precalculus Enhanced with Graphing Utilities

Question

Find the area of the blue shaded region in the figure, assuming

the quadrilateral inside the circle is a square.

Step-by-Step Solution

Answer

The area of the blue shaded region in the given figure is 41.04

Step 1. Given information

 A square inside the circle is given and Some parts of the regions are shaded.

We have to find out  the area of that shaded region

Step 2. Find radius and diameter

 According to the given information equation of the given circle is $x^2+y^2=36$and it   is the in the  form of standard equation of a circle at origin (0,0) and radius r $x^2+y^2=r^2$

   Hence radius is 6 and diagonal is 12 since the diameter is 2times of radius 

Step3. Finding the area of a square

Here radius is 6 and diagonal is 12.We know that diameter  of the circle is the diagonal of a square , thus diagonal of a square is 12

Thus area of a square is $$\begin{gathered} A =\frac{d^2}{2} ,d=diagonal \\ =\frac{{12}^2}{2} \\ =\frac{144}{2} \\ =72 \end{gathered}$$

Step 4. Finding the area of a circle

In the given circle radius r =6 , then area of circle

 $$\begin{gathered} A =\pi \times r^2 \\ =3.14\times 36 r=6 \\ =113.04 \end{gathered}$$

Step 5. Find the area of the blue shaded region

 The area of the circle here is 113.04 and the area of the square is 72.

 Then according to the given figure 

$$\begin{gathered} area of a blue shaded region = area of the circle -area of the square \\ =113.04-72 \\ =41.04 \end{gathered}$$