Question

How do you answer this question?

Answer 1

`148.91~~A`

Note that my solution requires the basic knowledge on sine and its inverse, arc sine.

Explanation:

Before we start, We sketch our situation:

This tells us that the radius of the circle is `10`cm.

Therefore, we can see that:

Now, the distance from the center to the downright vertex of the rectangle is `8`cm.

Using the Pythagoras triples, we see that at the bottom right portion of the rectangle, we form a `10-8-6` triangle.

Therefore, we say that the area of the triangle is `(8*6)/2` which is `24`cm squared.

We now have:

Now, let's figure out the angle `x` using the sine function.

Remember that `sinx="(opposite)/(hypotenuse)"`

Therefore, `sinx=6/10`

=>`sinx=3/5` solve for `x`.

=>`x=arcsin(3/5)`

=>`x~~36.87`(In degrees.)

Therefore, we have:

We can now figure out `y`.

Since `x` and `y` add up to a straight line, they are supplementary.

Therefore, `x+y=180`.

=>`36.87+y=180`

=>`y=143.13`

We can now find the area of the arc by using the fact that the area of an arc over the area of the circle it is in is equal to the angle of the arc over 360 degrees.

The area of the circle can be found using `A=pir^2`

=>`A=pi(10)^2`

=>`A=100pi`

We have:
`143.13/360=A/(100pi)`

=>`14313pi=360A`

=>`44965.91~~360A`

=>`124.91~~A`

We now add up the area of our `6-8-10` right triangle and our arc to get the final answer.

`124.91+24~~A`

=>`148.91~~A`