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16 Sep 2014, 00:13
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A semicircle with a diameter of \(\pi\) has a circle inscribed right in the middle, as shown below. What is the ratio of the area of the semicircle to the area that is not covered by the inscribed circle?
A. 4:1
B. 3:2
C. 3:1
D. 4:3
E. 2:1
A. 4:1
B. 3:2
C. 3:1
D. 4:3
E. 2:1
16 Sep 2014, 00:13
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Official Solution:
A semicircle with a diameter of \(\pi\) has a circle inscribed right in the middle, as shown below. What is the ratio of the area of the semicircle to the area that is not covered by the inscribed circle?
A. 4:1
B. 3:2
C. 3:1
D. 4:3
E. 2:1
We can observe that the radius of the large semicircle is twice that of the inscribed circle, with values \(\frac{\pi}{2}\) and \(\frac{\pi}{4}\), respectively. Since the area of a circle is proportional to the square of its radius, we know that the area of the large circle (which includes the semicircle) is four times greater than that of the inscribed circle. For instance, a circle with a radius of 2 has an area of \(4\pi\), which is four times greater than a circle with a radius of 1 and an area of \(\pi\).
As a result, the area of the semicircle is twice the area of the inscribed circle. Assuming that the area of the semicircle is 2 units^2, the area of the inscribed circle is 1 unit^2, and therefore, the area not covered by the inscribed circle is also 1 unit^2. Thus, the ratio of the area of the semicircle to the area not covered by the inscribed circle is \(\frac{2}{1}\).
Answer: E
A semicircle with a diameter of \(\pi\) has a circle inscribed right in the middle, as shown below. What is the ratio of the area of the semicircle to the area that is not covered by the inscribed circle?
A. 4:1
B. 3:2
C. 3:1
D. 4:3
E. 2:1
We can observe that the radius of the large semicircle is twice that of the inscribed circle, with values \(\frac{\pi}{2}\) and \(\frac{\pi}{4}\), respectively. Since the area of a circle is proportional to the square of its radius, we know that the area of the large circle (which includes the semicircle) is four times greater than that of the inscribed circle. For instance, a circle with a radius of 2 has an area of \(4\pi\), which is four times greater than a circle with a radius of 1 and an area of \(\pi\).
As a result, the area of the semicircle is twice the area of the inscribed circle. Assuming that the area of the semicircle is 2 units^2, the area of the inscribed circle is 1 unit^2, and therefore, the area not covered by the inscribed circle is also 1 unit^2. Thus, the ratio of the area of the semicircle to the area not covered by the inscribed circle is \(\frac{2}{1}\).
Answer: E
17 Feb 2016, 11:55
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I just plugged into the area formula. It takes a bit longer though.
Big half circle area = (π*(π/2)*(π/2) / 2 = π/8
Little circle area = π/16
Big Half circle - Little circle = area not covered
2/16-1/16 = 1/16
Ratio
1/8:1/16 = 2:1
Big half circle area = (π*(π/2)*(π/2) / 2 = π/8
Little circle area = π/16
Big Half circle - Little circle = area not covered
2/16-1/16 = 1/16
Ratio
1/8:1/16 = 2:1
