Mathematics: Find the area of the shaded region.

Find the area of the shaded region. - id660118, johnkings140

Ответ:

owlsandR5

12.03.2022

Check the picture below.

now, let's rationalize the denominator for "r" first.

also notice, that section is just 1/4 of the area of that circle, so, let's just find the area of the whole circle with that "r", get one-quarter of it, and subtract it from the half of the area of the square.

$\bf r=\cfrac{8}{\sqrt{2}}\cdot \cfrac{\sqrt{2}}{\sqrt{2}}\implies r=\cfrac{8\sqrt{2}}{2}\implies \boxed{r=4\sqrt{2}}
\\\\\\
\stackrel{\textit{area of that circle}}{A=\pi (4\sqrt{2})^2}\implies A=\pi (16\cdot 2)\implies A=32\pi 
\\\\\\
\textit{how much is one-quarter of that?}\quad \cfrac{32\pi }{4}\implies \boxed{8\pi }\\\\
-------------------------------\\\\$

$\bf \stackrel{\textit{area of the square}}{A=8\cdot 8}\implies A=64\qquad \stackrel{\textit{half of that square's area}}{32}\\\\
-------------------------------\\\\
\textit{shaded region}\implies \stackrel{\stackrel{\textit{half of that square}}{area}}{32}~~-~~\stackrel{\stackrel{\textit{one-quarter of the circle}}{area}}{8\pi }$

Find the area of the shaded region.

0,0(0 оценок)

Ответ:

eloqit3338

12.03.2022

Mathematics

-8π + 32

6.8675

Step-by-step explanation:

So we know that the are of the square is 64, and we know that the equation for the are of the shaded region is Area of square - 1/4 area of circle with a radius of 4√2 - area of 90/45/45 triangle with side lengths of 8 and 8.

s = square area

c = circle area

t = triangle area

∆ = shaded region area

∆ = s - 1/4c - t

input the numbers that we have for those variables:

∆ = 64 - 1/4(100.53) - 32

∆ = 64 - 25.1325 - 32

∆ = 32 - 25.1325

shade region approximate ≈ 6.8675

or to get an exact answer in terms of pi

∆ = 64 - 1/4(32π) - 32