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How do you solve this?

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Answer 1 · 2018-06-16T23:43:32

I tried this but check my maths...

Explanation:

Consider the Diagram:
enter image source here

I would use Trigonometry to find the sides of triangle AOC:

$\frac{r}{A C} = tan (30^{\circ})$ $r/(AC)=tan(30^@)$

$A C = \frac{3}{tan (30^{\circ})} = 5.2$ $AC=3/tan(30^@)=5.2$

and:

$r = O C sin (30^{\circ})$ $r=OCsin(30^@)$

$O C = \frac{3}{sin (30^{\circ})} = 6$ $OC=3/sin(30^@)=6$

the area of the triangle AOC will be:

$Area1 = \frac{A C \cdot r}{2} = 7.8 u.a.$ $"Area1"=(AC*r)/2=7.8 "u.a."$

we can multiply by $2$ $2$ to get the area of the two tringles:

$Area Triangles = 2 \cdot Area1 = 2 \cdot 7.8 = 15.6 u.a.$ $"Area Triangles"=2*"Area1"=2*7.8=15.6"u.a."$

We can subtract from this area the area of the circular sector AOB to get the violet area:

$Area Aector = \frac{1}{2} r^{2} \cdot angle in radians = \frac{1}{2} \cdot 3^{2} \cdot \frac{2}{3} π = 9.4 u.a.$ $"Area Aector"=1/2r^2*"angle in radians"=1/2*3^2*2/3pi=9.4"u.a."$

So the final area should be:

$Area shaded = 15.6 - 9.4 = 6.2 u.a.$ $"Area shaded"=15.6-9.4=6.2"u.a."$

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