Discussion
| Column A | Column B |
| The percent of the rectangle that is shaded | 25% |
| (A) | Column A is larger |
| (B) | ... |
| (C) | ... |
| (D) | ... |
| (E) | ... |
| (F) | ... |
The solution is
Posted: 02/18/2013 13:27
This is true if the base of the rectangle bisects along the diameter; what if it doesn't? Nothing specifies that the base of the rectangle = the diameter.
Posted: 02/26/2013 20:32
Scott, the problem says explicitly that it is a "semicircle inscribed in the rectangle", so it pretty much is established that the base of the rectangle is the diameter of the circle.
Posted: 04/03/2013 13:58
Hi I think it depends on the width and height of the rectangle. If for instance you pick 2 as the length and 1 and the width the answer. The column B will be larger.
However if you pick 6 as the length and 2 as the width. The rectangular area will be 12 (6*2), and the semicircle area will be ( pi*radious^2)/2 , or using the length 6 and width 3 ( pi*(3^2))/2. This will yield 4.5pi or 14.137 . Hence i think the answer should be D. There is no enough information.
However if you pick 6 as the length and 2 as the width. The rectangular area will be 12 (6*2), and the semicircle area will be ( pi*radious^2)/2 , or using the length 6 and width 3 ( pi*(3^2))/2. This will yield 4.5pi or 14.137 . Hence i think the answer should be D. There is no enough information.
Posted: 01/23/2014 13:26
The explanation says "calculating the percent of the rectangle that is shaded yields....25%." That would make column A ("percent of the rectangle that is shaded") and column B (25%) equal. In that case, the answer doesn't make sense. What am I missing here?
Posted: 01/23/2014 14:50
Lauren, that part of the explanation is just saying that the shaded area is less than (4-3)/4 or 1/4 or 25%.
Posted: 02/05/2014 05:45
Ok I see that now, thank you.
I do have another question. The solution says that "we can assume the dimensions are independent of the solution." How can we determine when the dimensions (or anything else) should be assumed vs realizing there is not enough information. For example, when it comes to triangles, we should not assume dimensions based on the drawing. I particularly remember a question where one angle of the triangle drawn was clearly larger than 90 and clearly obtuse (as in at least 120 degrees) but the solution told us we can't assume that it is obtuse and not acute. When do we know when to assume and when to not assume?
I do have another question. The solution says that "we can assume the dimensions are independent of the solution." How can we determine when the dimensions (or anything else) should be assumed vs realizing there is not enough information. For example, when it comes to triangles, we should not assume dimensions based on the drawing. I particularly remember a question where one angle of the triangle drawn was clearly larger than 90 and clearly obtuse (as in at least 120 degrees) but the solution told us we can't assume that it is obtuse and not acute. When do we know when to assume and when to not assume?
Posted: 06/19/2014 08:10
Reply: Lauren, sorry for the late reply. You already know the answer. You can assume when it is mathematically logical, such as, a circle is circumscribed inside a quadrangle; well, the quadrangle must be a square, etc.
Posted: 06/19/2014 06:40
I can't understand why the solution is B. If we take the radius of the circle to be r, then the shaded area is ((4-pi)r^2)/2, and this can be less, equal or more than 25%. Please show me where the mistake is, if there is any.
Posted: 06/19/2014 14:01
Hi Gui,
((4-pi)r^2)/2 is the area of the shaded region, but this is not a percentage (a fraction) of the area of the rectangle.
To form the percent, we divide the area of the shaded region by the area of the entire rectangle:
(Area of shaded region) / (Area of entire rectangle) =
[((4-pi)r^2)/2] / [4r^2] =
(4 - pi) / 2
This number is less than .25, or 25%.
Nova Press
((4-pi)r^2)/2 is the area of the shaded region, but this is not a percentage (a fraction) of the area of the rectangle.
To form the percent, we divide the area of the shaded region by the area of the entire rectangle:
(Area of shaded region) / (Area of entire rectangle) =
[((4-pi)r^2)/2] / [4r^2] =
(4 - pi) / 2
This number is less than .25, or 25%.
Nova Press
Posted: 03/10/2016 20:12
Just a note for a bug fix. I have an iPhone and many of your solutions don't show the full answer
If the length of the rectangle is 6, then the width would have to be 3, not 2.
If the width were 2, then the semicircle would not touch the top of the rectangle as it does in the given figure (it would rise above the rectangle because the radius of the semicircle would be 3). Hence, the semicircle would not be inscribed in the rectangle.
Nova Press