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a. Find the graph of their common region in the attachment
b. The area of the common region of the graphs is 8 units²
<h3>a. How to sketch the region common to the graphs?</h3>Since we have x ≥ 2, y ≥ 0, and x + y ≤ 6, we plot each graph separately and find their region of intersection.
- The graph of x ≥ 2 is the region right of the line x = 2.
- The graph of y ≥ 0 is the region above the line y = 0 or x-axis.
- To plot the graph of x + y ≤ 6, we first plot the graph of x + y = 6 ⇒ y = -x + 6. Then the graph of x + y ≤ 6 is the graph of y ≤ - x + 6.
So, the graph of x + y ≤ 6 is the region below the line y = - x + 6
From the graph, the regions intersect at (2, 0), (2, 4) and (6, 0)
Find the graph of their common region in the attachment
<h3>b. The area of the common region</h3>From the graph, we see that the common region is a right angled triangle with
- height = 4 units and
- base = 4 units
So, its area = 1/2 × height × base
= 1/2 × 4 units × 4 units
= 1/2 × 16 units²
= 8 units²
So, the area of the common region of the graphs is 8 units²
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