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The difference in the areas of circumscribing and inscribing circles of the region bounded by the curves y = -|x| + 12 andy = |x| + 4 will be:
- a)4π
- b)8π
- c)10π
- d)12π
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The difference in the areas of circumscribing and inscribing circles o...
As shown in the graph, we get the intersection points (-4, 8), (4, 8), (0, 12) and (0, 4).
The enclosed region would be a square with each side of length of 4√2
The radius of the inscribed circle = 2√2
It's area = 8π
The radius of the circle circumscribing the square will be 4.
It’s area = 16π
The difference between these two areas = 16π - 8π = 8π
Hence, option 2.
Most Upvoted Answer
The difference in the areas of circumscribing and inscribing circles o...
As shown in the graph, we get the intersection points (-4, 8), (4, 8), (0, 12) and (0, 4).
The enclosed region would be a square with each side of length of 4√2
The radius of the inscribed circle = 2√2
It's area = 8π
The radius of the circle circumscribing the square will be 4.
It’s area = 16π
The difference between these two areas = 16π - 8π = 8π
Hence, option 2.
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Community Answer
The difference in the areas of circumscribing and inscribing circles o...
First, let's find the points of intersection of the two curves.
For y = -|x| + 12, we have:
-x + 12 = |x|
If x ≥ 0, then -x + 12 = x, which gives x = 6.
If x < 0,="" then="" -x="" +="" 12="-x," which="" gives="" x="0." />
So, the points of intersection for this curve are (6, 6) and (0, 12).
For y = |x| + 4, we have:
x + 4 = |x|
If x ≥ 0, then x + 4 = x, which gives x = -4.
If x < 0,="" then="" x="" +="" 4="-x," which="" gives="" x="-2." />
So, the points of intersection for this curve are (-4, 0) and (-2, 2).
Now, let's find the radii of the inscribing and circumscribing circles of the region bounded by the curves.
For the inscribing circle, the radius is the distance from the origin to the closest point on the curve y = -|x| + 12. The closest point to the origin is (0, 12), so the radius is 12.
For the circumscribing circle, the radius is the distance from the origin to the farthest point on the curve y = |x| + 4. The farthest point to the origin is (6, 10), so the radius is 10.
The area of a circle is given by A = πr^2.
For the inscribing circle, the area is A_ins = π(12)^2 = 144π.
For the circumscribing circle, the area is A_circ = π(10)^2 = 100π.
The difference in the areas of the two circles is A_circ - A_ins = 100π - 144π = -44π.
So, the answer is -44π.
For y = -|x| + 12, we have:
-x + 12 = |x|
If x ≥ 0, then -x + 12 = x, which gives x = 6.
If x < 0,="" then="" -x="" +="" 12="-x," which="" gives="" x="0." />
So, the points of intersection for this curve are (6, 6) and (0, 12).
For y = |x| + 4, we have:
x + 4 = |x|
If x ≥ 0, then x + 4 = x, which gives x = -4.
If x < 0,="" then="" x="" +="" 4="-x," which="" gives="" x="-2." />
So, the points of intersection for this curve are (-4, 0) and (-2, 2).
Now, let's find the radii of the inscribing and circumscribing circles of the region bounded by the curves.
For the inscribing circle, the radius is the distance from the origin to the closest point on the curve y = -|x| + 12. The closest point to the origin is (0, 12), so the radius is 12.
For the circumscribing circle, the radius is the distance from the origin to the farthest point on the curve y = |x| + 4. The farthest point to the origin is (6, 10), so the radius is 10.
The area of a circle is given by A = πr^2.
For the inscribing circle, the area is A_ins = π(12)^2 = 144π.
For the circumscribing circle, the area is A_circ = π(10)^2 = 100π.
The difference in the areas of the two circles is A_circ - A_ins = 100π - 144π = -44π.
So, the answer is -44π.
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The difference in the areas of circumscribing and inscribing circles of the region bounded by the curves y = -|x| + 12 andy = |x| + 4 will be:a)4πb)8πc)10πd)12πCorrect answer is option 'B'. Can you explain this answer?
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