Question

Find the area of the shaded region.
Round to the nearest tenth.

Answer 1

The area of the shaded region is 11.606 square centimeters.

Explanation:

The area of the shaded region is obtained by subtracting the area of the triangle from the area of the circular section. The area of the triangle (
`A_(t)`), in square centimeters, can be calculated by the Heron's formula:

`A_(t) = √(s\cdot (s-a)\cdot (s-b)\cdot (s-c))` (1)

`s = (a + b + c)/(2)` (2)

Where:

`a`,
`b`,
`c` - Lengths of the sides of the triangle, in centimeters.

`s` - Semiperimeter, in centimeters.

If we know that
`a = 10.50\,cm` and
`b = c = 9.28\,cm`, then the area of the triangle is:

`s = (10.50\,cm + 2\cdot (9.28\,cm))/(2)`

`s = 14.53\,cm`

`A_(t) = \sqrt{(14.53\,cm)\cdot (14.53\,cm - 10.50\,cm)\cdot (14.53\,cm - 9.28\,cm)^(2)}`

`A_(t) \approx 40.174\,cm^(2)`

And the area of the circular section (
`A_(c)`), in square centimeters, is determined by the following formula:

`A_(c) = \left((\theta\cdot \pi)/(360) \right)\cdot r^(2)` (3)

Where:

`r` - Radius of the circle, in centimeters.

`\theta` - Internal angle, in sexagesimal degrees.

If we know that
`r = 9.28\,cm` and
`\theta = 68.9^(\circ)`, then the area of the circular section is:

`A_(c) = \left((68.9)/(360)\right)\cdot \pi\cdot (9.28\,cm)^(2)`

`A_(c) \approx 51.780\,cm^(2)`

Finally, the area of the shaded region (
`A`), in square centimeters, is:

`A = A_(c) - A_(t)` (4)

`A = 51.780\,cm^(2)- 40.174\,cm^(2)`

`A = 11.606\,cm^(2)`

The area of the shaded region is 11.606 square centimeters.