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14 Sep 2019, 11:36
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
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Question Stats:
50% (02:53) correct
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Given that a straight line in the xy-plane passes through the point (3,48) in the first quadrant and intersects the positive x- and y-axes at points P and Q respectively. Let O designate the origin. Find the minimum value of OP×OQ.
A. 48
B. 96
C. 198
D. 576
E. 624
A. 48
B. 96
C. 198
D. 576
E. 624
14 Sep 2019, 11:46
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nick1816
Given:
1. Given that a straight line in the xy-plane passes through the point (3,48) in the first quadrant and intersects the positive x- and y-axes at points P and Q respectively.
2. Let O designate the origin.
Asked: Find the minimum value of OP×OQ.
Let the equation of the line l be
x/a + y/b = 1
where a, b > 0
3/a + 48/b = 1
3b + 48a = ab
Minimum value of OP * OQ = min(ab) = 1/ Max (1/a*1/b)
3/a + 48/b = 1
Sum is fixed; product is maximum when terms are equal
3/a = 48/b
48a = 3b
b = 16a
48a + 48a = 16a^2 = 96a
a = 96/16 = 6
b = 96
Minimum value of OP * OQ = 6*96 = 576
IMO D
23 Sep 2020, 07:04
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I approached the question from the perspective of trying to logically analyze each Answer Choice.
1st). Visualizing the Line and figuring out what the question is asking
For the Line to pass through Point (3 , 48) AND cross both the +Pos. X-Axis and +Pos. Y-Axis, the Line must be a downward sloping line
Further, Point P is the X intercept of the line and Point Q is the Y Intercept of the line. What the question is essentially asking us to do is MINIMIZE the Product of the (X Intercept Value) * (Y Intercept Value) = ?
The 1st thing to notice is that since the Downward Sloping Line crosses through Point (3 , 48), the point at which the line crosses the Y Axis MUST be greater than > +48. The Y-Intercept can NOT be at Q = (0 , 48).
Further, if the line WERE to cross the X-Axis at P = (3, 0), the only way the line would intersect Point (3 , 48) is if the line was a vertical line with an equation of: X = 3
However, this is not possible because the vertical line X = 3 will never cross the (+)Pos. Y Axis at Point Q. Also, we already established that the Line must be Downward Sloping and that the Y Intercept at Point Q must be greater than > 48
Therefore, the X-Intercept at Point P must be greater than > 3
From these observations, we can say the following:
(OP) * (PQ) > (48) * (3)
(OP) * (PQ) > 144
This eliminates answers A and B right off the bat.
For Answer C = 198:
There are 12 Total Factors of 198, which gives us 6 (+) Pos. Factor Pairs of Integers that multiply to 198.
While the problem does not say we must have integer values of P and Q, it is a good place to start to see which values are possible given the constraints we found.
The 6 (+) Pos. Factor Pairs that multiply = 198 are:
(1 * 198) —- (2 * 99) — (3 * 66) —-(6 * 33) —-(9 * 22) —-(11 * 18)
Given that P >48 and Q >3, none of this Factor Pairs can be made to Satisfy both constraints at the same time and also multiply to 198
Furthermore, the sqrt(198) is a little bit greater than 14. As we keep spreading the 2 factors apart, logically there is no way we can find 2 (+)positive factors for P and Q that would satisfy both conditions AND multiply to 198.
This leaves us with D and E. At this point, we could make a guess and move on if we are short on time.
However, analyzing Answer D - 576, you can quickly spot that 6 is a factor of 576 —— (6 * 96 = 576)
Let the X- Intercept be at the Coordinates (6 , 0).
Given that the Line passes through Point (3 , 48), will the Line have a Y Intercept = Q = 96 when the X-Intercept = 6——- leading to the fact that (OP) * (OQ) = 576?
1st) Find the Equation of this Line with X Intercept (6 , 0):
We have 2 points to quickly find the slope: (6 , 0) and (3 , 48)
m = (rise) / (run) = (48 - 0) / (3 - 6) = -16
Slope Intercept Equation: y = - 16x + b
***where b = Y-Intercept at Point Q
2nd) Plug Point (3 , 48) into the slope intercept equation above in order to find b:
48 = -16*(3) + b
b = 96 ——- works! The Line with Slope -16 that passes through (3 , 48) and (6 ,0) will have a Y-Intercept = b = 96
3rd) In conclusion: Given that the Line must pass through (3 , 48) and cross both the (+)Pos. Y-Axis and (+) Pos. X-Axis ——
The X Intercept at Point P would have the coordinates (6 , 0). Line OP would = 6 units
The Y Intercept at Point Q would have the coordinates (0 , 96). Line OQ would = 96 units
OP * OQ = 6 * 96 = 576
Thus, given the Answer Choices, Choice D = 576 is the MIN Value that actually works given the constraints.
Answer -D-
EDIT: the post above mine is most likely the preferred way to do this problem. However, the knowledge required is high level and this is an alternate way to get there by Testing the Answer Choices.
Posted from my mobile device
1st). Visualizing the Line and figuring out what the question is asking
For the Line to pass through Point (3 , 48) AND cross both the +Pos. X-Axis and +Pos. Y-Axis, the Line must be a downward sloping line
Further, Point P is the X intercept of the line and Point Q is the Y Intercept of the line. What the question is essentially asking us to do is MINIMIZE the Product of the (X Intercept Value) * (Y Intercept Value) = ?
The 1st thing to notice is that since the Downward Sloping Line crosses through Point (3 , 48), the point at which the line crosses the Y Axis MUST be greater than > +48. The Y-Intercept can NOT be at Q = (0 , 48).
Further, if the line WERE to cross the X-Axis at P = (3, 0), the only way the line would intersect Point (3 , 48) is if the line was a vertical line with an equation of: X = 3
However, this is not possible because the vertical line X = 3 will never cross the (+)Pos. Y Axis at Point Q. Also, we already established that the Line must be Downward Sloping and that the Y Intercept at Point Q must be greater than > 48
Therefore, the X-Intercept at Point P must be greater than > 3
From these observations, we can say the following:
(OP) * (PQ) > (48) * (3)
(OP) * (PQ) > 144
This eliminates answers A and B right off the bat.
For Answer C = 198:
There are 12 Total Factors of 198, which gives us 6 (+) Pos. Factor Pairs of Integers that multiply to 198.
While the problem does not say we must have integer values of P and Q, it is a good place to start to see which values are possible given the constraints we found.
The 6 (+) Pos. Factor Pairs that multiply = 198 are:
(1 * 198) —- (2 * 99) — (3 * 66) —-(6 * 33) —-(9 * 22) —-(11 * 18)
Given that P >48 and Q >3, none of this Factor Pairs can be made to Satisfy both constraints at the same time and also multiply to 198
Furthermore, the sqrt(198) is a little bit greater than 14. As we keep spreading the 2 factors apart, logically there is no way we can find 2 (+)positive factors for P and Q that would satisfy both conditions AND multiply to 198.
This leaves us with D and E. At this point, we could make a guess and move on if we are short on time.
However, analyzing Answer D - 576, you can quickly spot that 6 is a factor of 576 —— (6 * 96 = 576)
Let the X- Intercept be at the Coordinates (6 , 0).
Given that the Line passes through Point (3 , 48), will the Line have a Y Intercept = Q = 96 when the X-Intercept = 6——- leading to the fact that (OP) * (OQ) = 576?
1st) Find the Equation of this Line with X Intercept (6 , 0):
We have 2 points to quickly find the slope: (6 , 0) and (3 , 48)
m = (rise) / (run) = (48 - 0) / (3 - 6) = -16
Slope Intercept Equation: y = - 16x + b
***where b = Y-Intercept at Point Q
2nd) Plug Point (3 , 48) into the slope intercept equation above in order to find b:
48 = -16*(3) + b
b = 96 ——- works! The Line with Slope -16 that passes through (3 , 48) and (6 ,0) will have a Y-Intercept = b = 96
3rd) In conclusion: Given that the Line must pass through (3 , 48) and cross both the (+)Pos. Y-Axis and (+) Pos. X-Axis ——
The X Intercept at Point P would have the coordinates (6 , 0). Line OP would = 6 units
The Y Intercept at Point Q would have the coordinates (0 , 96). Line OQ would = 96 units
OP * OQ = 6 * 96 = 576
Thus, given the Answer Choices, Choice D = 576 is the MIN Value that actually works given the constraints.
Answer -D-
EDIT: the post above mine is most likely the preferred way to do this problem. However, the knowledge required is high level and this is an alternate way to get there by Testing the Answer Choices.
Posted from my mobile device
