find the total area of the rectangles in the figure below, where the equation of the line is f(x) = x - 11.

Explanation:

Step1: Determine the base and height of each rectangle

The base of each rectangle is 1 (since the x - values increase by 1 unit between consecutive rectangles). For the first rectangle at (x = 12), (y_1=f(12)=12 - 11=1). For the second rectangle at (x = 13), (y_2=f(13)=13 - 11 = 2). For the third rectangle at (x = 14), (y_3=f(14)=14 - 11=3). For the fourth rectangle at (x = 15), (y_4=f(15)=15 - 11 = 4).

Step2: Use the area formula for a rectangle

The area of a rectangle is (A = b\times h), where (b) is the base and (h) is the height. The total area (A_{total}) of the rectangles is the sum of the areas of each rectangle. Since (b = 1) for each rectangle, (A_{total}=1\times1+1\times2 + 1\times3+1\times4).

Step3: Calculate the sum

[ \begin{align*} A_{total}&=1 + 2+3 + 4\ &=\frac{4\times(4 + 1)}{2}\ &=10 \end{align*} ]

Answer:

10