26. In the rectangle shown above, AD=6, AB=8. What is the probability that PD<45^1/2?
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Q 26
Source: Beat The GMAT — Problem Solving |
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manpsingh87
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AD=6; AP=8-x;engg.manik wrote:26. In the rectangle shown above, AD=6, AB=8. What is the probability that PD<45^1/2?
by using Pythagoras theorem we have; PD^2=6^2+(8-x)^2;
PD^2=36+64+x^2-16x;
PD=sqrt(x^2-16x+100);
now x can vary from 1 to 7;
and the values for which above expression will fulfill the desired condition x=5,6,7; hence required probability is 3/7...!!!
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I'm getting 1/3. Perhaps I'm reading the problem wrong.
AD = 6 , AP = x
6^2 + x^2 < 45
only three options that work for this are 0, 1 and 2, since plugging in 3 = 45 and doesn't satisfy the inequality.
other options are 3-8. Total of 9 options.
I've assumed that having AP = 0 is allowable.
AD = 6 , AP = x
6^2 + x^2 < 45
only three options that work for this are 0, 1 and 2, since plugging in 3 = 45 and doesn't satisfy the inequality.
other options are 3-8. Total of 9 options.
I've assumed that having AP = 0 is allowable.
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Ian Stewart
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The wording of the question doesn't make sense. It asks for a probability, but if we're finding a probability, we have to be selecting something from a clearly defined set. The question doesn't even mention selecting anything, let alone mention how we're making the selection. I think it should read: "If point P is a randomly selected pointed on line segment AB, what is the probability the length of PD is less than √45?"
This is then a 1-dimensional geometric probability question. In general, if we choose a point at random from a line of length L, and we want to find the probability that point will be in some smaller line segment of length d, the probability will just be d/L. We often do this kind of calculation specifically on the number line -- if, say you pick x at random so that 0 < x < 5, and you want to know the probability that 0 < x < 2, the answer is just 2/5 -- but the same principle applies to a random selection from any line.
Now, returning to the question, if PD < √45, then by the Pythagorean Theorem, 6^2 + PA^2 < (√45)^2, so PA^2 < 45 - 36, and PA < 3. So the length of the line PA from which we can choose P is 3, and since the total length of the line from which we can choose P is 8, the answer is 3/8.
This is then a 1-dimensional geometric probability question. In general, if we choose a point at random from a line of length L, and we want to find the probability that point will be in some smaller line segment of length d, the probability will just be d/L. We often do this kind of calculation specifically on the number line -- if, say you pick x at random so that 0 < x < 5, and you want to know the probability that 0 < x < 2, the answer is just 2/5 -- but the same principle applies to a random selection from any line.
Now, returning to the question, if PD < √45, then by the Pythagorean Theorem, 6^2 + PA^2 < (√45)^2, so PA^2 < 45 - 36, and PA < 3. So the length of the line PA from which we can choose P is 3, and since the total length of the line from which we can choose P is 8, the answer is 3/8.
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