Answer: To find the area of the top surface of the wedge, we'll first find the area of the corresponding sector of the circle, and then subtract the area of the triangle formed by the radius and the central angle.
Area of the sector:
The formula to find the area of a sector of a circle is given by:
Area of sector = (Central angle / 2π) * π * r^2
where:
Central angle = π/2 radians (given)
r = 6 inches (radius)
Area of sector = (π/2π) * π * 6^2
= (1/2) * π * 36
= 18π square inches
Area of the triangle:
The triangle is formed by the radius (6 inches) and the central angle (π/2 radians). The area of a triangle can be found using the formula:
Area of triangle = (1/2) * base * height
In this case, the base and height are both 6 inches (since the triangle is isosceles).
Area of triangle = (1/2) * 6 * 6
= 18 square inches
Area of the top surface of the wedge:
To find the area of the top surface of the wedge, we subtract the area of the triangle from the area of the sector:
Area of top surface = Area of sector - Area of triangle
= 18π square inches - 18 square inches
= (18π - 18) square inches
= 18(π - 1) square inches
So, the area of the top surface of the wedge is 18(π - 1) square inches.