Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
To solve this problem, let's break it down step by step.
1. Identify the coordinates and lengths of segments:
- The entire line segment [tex]\( JM \)[/tex] has endpoints at 0 and 25. Hence, the total length of [tex]\( JM \)[/tex] is:
[tex]\[ JM = 25 - 0 = 25 \][/tex]
- Points [tex]\( K \)[/tex] and [tex]\( L \)[/tex] have coordinates 5 and 12, respectively. Thus, the length of segment [tex]\( JL \)[/tex] is:
[tex]\[ JL = 12 - 0 = 12 \][/tex]
- The length of segment [tex]\( KL \)[/tex] (since [tex]\( K \)[/tex] is at 5 and [tex]\( L \)[/tex] is at 12) is:
[tex]\[ KL = 12 - 5 = 7 \][/tex]
2. Calculate probabilities:
- The probability that a point placed on [tex]\( JM \)[/tex] falls within segment [tex]\( JL \)[/tex] is the length of [tex]\( JL \)[/tex] over the total length of [tex]\( JM \)[/tex]:
[tex]\[ \text{Probability of JL} = \frac{JL}{JM} = \frac{12}{25} = 0.48 \][/tex]
- The probability that a second point falls outside the segment [tex]\( KL \)[/tex] (i.e., on [tex]\( JM \)[/tex] but not on [tex]\( KL \)[/tex]) can be found by subtracting the length of [tex]\( KL \)[/tex] from the total length of [tex]\( JM \)[/tex], and then dividing by the total length of [tex]\( JM \)[/tex]:
[tex]\[ \text{Probability of not KL} = \frac{JM - KL}{JM} = \frac{25 - 7}{25} = \frac{18}{25} = 0.72 \][/tex]
3. Calculate combined probability:
- The combined probability that the first point is placed on [tex]\( JL \)[/tex] and the second point is not placed on [tex]\( KL \)[/tex] is the product of the two individual probabilities:
[tex]\[ \text{Combined probability} = 0.48 \times 0.72 = 0.3456 \][/tex]
4. Convert probability to a fraction:
- To express this probability as a fraction of the total possible number segments (since [tex]\( JM \)[/tex] is discretized), multiply by [tex]\( 25^2 \)[/tex] (the square of the total length, as we're considering two points on the segment):
[tex]\[ \text{Fraction solution: } 0.3456 \times 25^25 = 0.3456 \times 625 = 216 \][/tex]
Thus, the probability that a point on [tex]\( JM \)[/tex] is placed first on [tex]\( JL \)[/tex] and a second point is not placed on [tex]\( KL \)[/tex] is:
[tex]\[ \frac{216}{625} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{216}{625}} \][/tex]
1. Identify the coordinates and lengths of segments:
- The entire line segment [tex]\( JM \)[/tex] has endpoints at 0 and 25. Hence, the total length of [tex]\( JM \)[/tex] is:
[tex]\[ JM = 25 - 0 = 25 \][/tex]
- Points [tex]\( K \)[/tex] and [tex]\( L \)[/tex] have coordinates 5 and 12, respectively. Thus, the length of segment [tex]\( JL \)[/tex] is:
[tex]\[ JL = 12 - 0 = 12 \][/tex]
- The length of segment [tex]\( KL \)[/tex] (since [tex]\( K \)[/tex] is at 5 and [tex]\( L \)[/tex] is at 12) is:
[tex]\[ KL = 12 - 5 = 7 \][/tex]
2. Calculate probabilities:
- The probability that a point placed on [tex]\( JM \)[/tex] falls within segment [tex]\( JL \)[/tex] is the length of [tex]\( JL \)[/tex] over the total length of [tex]\( JM \)[/tex]:
[tex]\[ \text{Probability of JL} = \frac{JL}{JM} = \frac{12}{25} = 0.48 \][/tex]
- The probability that a second point falls outside the segment [tex]\( KL \)[/tex] (i.e., on [tex]\( JM \)[/tex] but not on [tex]\( KL \)[/tex]) can be found by subtracting the length of [tex]\( KL \)[/tex] from the total length of [tex]\( JM \)[/tex], and then dividing by the total length of [tex]\( JM \)[/tex]:
[tex]\[ \text{Probability of not KL} = \frac{JM - KL}{JM} = \frac{25 - 7}{25} = \frac{18}{25} = 0.72 \][/tex]
3. Calculate combined probability:
- The combined probability that the first point is placed on [tex]\( JL \)[/tex] and the second point is not placed on [tex]\( KL \)[/tex] is the product of the two individual probabilities:
[tex]\[ \text{Combined probability} = 0.48 \times 0.72 = 0.3456 \][/tex]
4. Convert probability to a fraction:
- To express this probability as a fraction of the total possible number segments (since [tex]\( JM \)[/tex] is discretized), multiply by [tex]\( 25^2 \)[/tex] (the square of the total length, as we're considering two points on the segment):
[tex]\[ \text{Fraction solution: } 0.3456 \times 25^25 = 0.3456 \times 625 = 216 \][/tex]
Thus, the probability that a point on [tex]\( JM \)[/tex] is placed first on [tex]\( JL \)[/tex] and a second point is not placed on [tex]\( KL \)[/tex] is:
[tex]\[ \frac{216}{625} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{216}{625}} \][/tex]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.