Question

Please i need help ASAP pls explain your answer

Answer 1

Answer:

$r=5.79898987...\\A=510.31110884...$

Step-by-step explanation:

Challenging question!

Start by constructing a square in the lower-left quadrant of the circle (that isn't shaded) that is tangent to center C, OP, and OQ:

(See picture)

With the blue line constructed, we can conclude that the blue line (the blue line is the measure from the center of the inscribed circle to the center of the quarter circle) measures $\sqrt{2} r$ due to the Pythagorean theorem.

The orange line is the radius of the inscribed circle, so adding the blue line and the orange line gives the radius of the quarter circle, so:

$\sqrt{2}r +r=14$

Solve the equation with a bit of algebra to get:

$\sqrt{2} r+r=14\\r(\sqrt{2}+1)=14\\r=\frac{14}{\sqrt{2}+1 }\\r=5.79898987...$

Now, we will find the area of the inscribed circle:

$A=\pi r^2\\A=\frac{22}{7} *(5.79898987...)^2\\A=105.68889115...$

And we will now find the area of the quarter circle...:

$A_{2} =\pi r^2\\A_{2}=\frac{22}{7}*14^2\\A_2=616$

...and we subtract both results...:

$A_{3}= 616-105.68889115...\\A_3=510.31110884...$

To get the area of the shaded region.

Not sure what does the last part ("correct to 3 s. f.") meant but that's all I can do.