The values of r for which the shaded region will be greater than or equal to 12 square units is;
r ≥ 8
We can see a rectangle and a triangle in the given image.
With the triangle being the unshaded portion of the rectangle.
Thus;
Area of shaded portion = Area of rectangle - area of rectangle.
We know that formula for area of rectangle is;
A_r = length × width
Thus; A_r = 3 × r
A_r = 3r
Area of a triangle is;
A_t = ½ × base × height
A_t = ½ × r × 3
A_t = ½(3r)
Thus;
Area of shaded portion = 3r - ½(3r)
Area of unshaded portion = 1.5r
We are told that the area of the unshaded region should be greater than or equal to 12. Thus;
1.5r ≥ 12
r ≥ 12/1.5
r ≥ 8
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