The leng of the smaller rectangle is 1 inch less than twice its wide. both the dimensions of the larger rectangle are 2 inches longer than the smaller rectangle. the area of the shaded region is 86 square inches. what is the area of the smaller rectangle?
assuming the smaller rectangle is enclosed by the larger, we have
small rectangle: w(2w-1)
large rectangle: (w+2)(2w-1+2)
Assuming the shaded region is the difference between the two rectangles, we have
(w+2)(2w+1)-w(2w-1) = 86
w = 14
So, the smaller is 14x27, with area=378
To find the area of the smaller rectangle, we need to determine its dimensions first. Let's go step by step.
Let's assume the width of the smaller rectangle is x inches.
According to the statement, the length of the smaller rectangle is 1 inch less than twice its width. So, the length would be (2x - 1) inches.
The dimensions of the larger rectangle are 2 inches longer than those of the smaller rectangle. Hence, the width of the larger rectangle would be (x + 2) inches, and the length would be (2x - 1 + 2) = (2x + 1) inches.
The area of a rectangle is calculated by multiplying its length by its width. Therefore, the area of the larger rectangle is:
Area of larger rectangle = (x + 2) * (2x + 1)
Now, let's consider the shaded region. The shaded region is the difference between the area of the larger rectangle and the area of the smaller rectangle. We know that the area of the shaded region is 86 square inches.
Area of shaded region = Area of larger rectangle - Area of smaller rectangle
86 = (x + 2) * (2x + 1) - x * (2x - 1)
To find x, we can simplify the equation and solve the quadratic equation for x. Then, substitute the value of x into the expression for the smaller rectangle's area.
However, at this point, we need additional information or an equation to solve for x.