Solve for the area when a rectangle of length 4cm and width 3cm has one vertex at the origin of a system of axes and another vertex on a quarter of a circle whose center is also at the origin. Find the shaded area?
a hexagonal prism 6ft tall with a regular base measuring 9fton each and an apothem of lenght7.8ft
can't see the shaded area, but clearly the circle has radius 5.
To solve this problem and find the shaded area, we need to break it down into smaller steps:
Step 1: Find the area of the rectangle
The formula to calculate the area of a rectangle is given by:
Area = Length * Width
Given that the length of the rectangle is 4cm and the width is 3cm, we can find the area:
Area = 4cm * 3cm = 12cm^2
Step 2: Find the area of the quarter circle
The formula to calculate the area of a quarter circle is given by:
Area of quarter circle = (π * r^2) / 4
Since the center of the quarter circle is at the origin, and one vertex of the rectangle lies on the quarter circle, the distance from the origin to that vertex equals the radius of the quarter circle.
The distance from the origin to the vertex can be calculated using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Where (x1, y1) is the coordinate of the origin (0, 0) and (x2, y2) is the coordinate of the vertex on the rectangle. In this case, the vertex lies on the x-axis, so its coordinates are (4, 0).
Distance = √((4 - 0)^2 + (0 - 0)^2) = 4cm
Now, we can substitute the radius value into the area formula for the quarter circle:
Area of quarter circle = (π * (4cm)^2) / 4 = 4π cm^2
Step 3: Find the shaded area
Since the shaded area is the difference between the area of the rectangle and the area of the quarter circle, we can subtract the latter from the former:
Shaded area = Area of rectangle - Area of quarter circle
Shaded area = 12cm^2 - 4π cm^2
Therefore, the shaded area is 12cm^2 - 4π cm^2.