What is the approximate area of Region \(Q\) in square units?

[M7MBA]

Region \(Q,\) shown in the figure, is defined by

\((x-6)^2+(y-4)^2 \le 100,\) with \(y\ge 0,\) and \(x\ge 0.\)

What is the approximate area of Region \(Q\) in square units?

A. between 75 and 125
B. between 125 and 175
C. between 175 and 225
D. between 225 and 275
E. between 275 and 325


[spoiler]OA=C[/spoiler]

Source: Magoosh

[Scott@TargetTestPrep]

Capture.PNG

Region \(Q,\) shown in the figure, is defined by

\((x-6)^2+(y-4)^2 \le 100,\) with \(y\ge 0,\) and \(x\ge 0.\)

What is the approximate area of Region \(Q\) in square units?

A. between 75 and 125
B. between 125 and 175
C. between 175 and 225
D. between 225 and 275
E. between 275 and 325


[spoiler]OA=C[/spoiler]

Source: Magoosh

Solution:

We see that the point (12, 12) is a point on the circle since (12 - 6)^2 + (12 - 4)^2 = 6^2 + 8^2 = 100. Now we can draw a square with side length of 12 in the diagram below (the two red sides and the coordinate axes) as a lower bound for the area of Region Q (i.e., the shaded region).

Similarly, we can draw a rectangle with length of 16 and width of 14 (the two orange sides and the coordinate axes) as an upper bound for the area of Region Q. Therefore, a lower bound for the area of Region Q is 12^2 = 144 and an upper bound is 14 x 16 = 224. In other words, 144 < Area of Region Q < 224. This makes both choices B and C as a possible answer.

We see that the triangle on the top of the square has an area of ½ x 12 x 2 = 12 and the triangle on the right side of the square has an area of ½ x 12 x 4 = 24. Therefore, we can improve the lower bound for the area of Region Q to be 144 + 12 + 24 = 180. In other words, we can say now that 180 < Area of Region Q < 224, which makes choice C the only correct answer.

Answer: C