95.6k views
4 votes
Given that the curves are formed from quarter circles, find the area of the shaded region. Give your answer in terms of T. 12m 12m​

Given that the curves are formed from quarter circles, find the area of the shaded-example-1
User Srikar
by
8.5k points

2 Answers

3 votes

the area of the shaded region:


= 2(the \ area \ of \ a \ quarter \ circle) - (the \ area \ of \ the \ square)\\


= 2\Big((\pi 12^(2) )/(4)\Big) - 12^(2) = (144 \pi)/(2) - 144 = 72 \pi - 144\\


= 72 (\pi - 2) \ m^(2)

User RoundOutTooSoon
by
9.1k points
5 votes

Answer:

72π-144 m²

Explanation:

You want the area of a shaded region consisting of the overlap of two quarter circles in a 12 m square.

Segments

If we draw a diagonal from upper left to lower right through the figure, the shaded area is divided into two 90° segments of a circle of radius 12 m.

The formula for the area of a segment is ...

A = 1/2r²(θ -sin(θ))

where θ is the measure of the central angle.

For θ = π/2 radians, this is ...

A = 1/2r²(π/2 -1) . . . . . half the shaded area

Shaded area

Then the whole shaded area is ...

2 × 1/2r²(π/2 -1) = (12 m)²(π/2 -1) = 72π -144 m²

The area of the shaded region is 72π -144 m².

__

Additional comment

If we expand the shaded area formula, we get ...

A =1/2πr² -r²

This is recognizable as twice the area of a quarter circle, less the area of the square.

<95141404393>

User Kutschenator
by
9.0k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories