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Sagot :
Let's break down the problem step by step to find the correct answer and identify Omyra's errors.
1. Determine the distance Joseph and Isabelle traveled:
- Joseph jogged north at a speed of 8 km/h for 1.5 hours.
[tex]\[ \text{Distance traveled by Joseph} = 8 \text{ km/h} \times 1.5 \text{ hours} = 12 \text{ km} \][/tex]
- Isabelle rode west at a speed of 12 km/h for 1.5 hours.
[tex]\[ \text{Distance traveled by Isabelle} = 12 \text{ km/h} \times 1.5 \text{ hours} = 18 \text{ km} \][/tex]
2. Use the Pythagorean theorem to find the distance between Joseph and Isabelle:
Since Joseph and Isabelle are moving perpendicular to each other (one north, one west), we apply the Pythagorean theorem:
[tex]\[ d^2 = (\text{Distance of Joseph})^2 + (\text{Distance of Isabelle})^2 \][/tex]
[tex]\[ d^2 = 12^2 + 18^2 \][/tex]
3. Calculate the squares:
[tex]\[ 12^2 = 144 \][/tex]
[tex]\[ 18^2 = 324 \][/tex]
4. Sum the squares:
[tex]\[ 144 + 324 = 468 \][/tex]
5. Calculate the square root to find the distance [tex]\(d\)[/tex]:
[tex]\[ d = \sqrt{468} \approx 21.63 \text{ km} \][/tex]
With these steps, we found the distance between Joseph and Isabelle to be approximately 21.63 km. Now let's identify Omyra's errors based on her work and the correct calculations above.
Omyra's work:
[tex]\[ 8^2 + 12^2 = d^2 \][/tex]
[tex]\[ 64 + 24 = d^2 \][/tex]
[tex]\[ 88 = d^2 \][/tex]
Errors in Omyra's calculations:
1. Error in squaring the speeds:
Omyra correctly squared the speeds initially in her work as follows:
- [tex]\(8^2 = 64\)[/tex]
- However, she incorrectly noted [tex]\(12^2 = 24\)[/tex] whereas it should be [tex]\(12^2 = 144\)[/tex].
2. Error in summing the squares:
The correctly squared values of distances should be:
[tex]\[ 64 + 144 = 208 \][/tex]
But Omyra incorrectly added:
[tex]\[ 64 + 24 = 88 \][/tex]
3. Error in calculating the final distance:
Omyra incorrectly used the sum [tex]\(d^2 = 88\)[/tex], leading to an incorrect final distance calculation. The correct step should be finding the square root of the correct sum:
[tex]\[ d = \sqrt{208} \approx 14.42 \text{ km} \][/tex]
Correct statements describing her errors:
1. Omyra incorrectly calculated [tex]\(12^2 \)[/tex] as [tex]\(24\)[/tex] instead of [tex]\(144\)[/tex].
2. Omyra incorrectly summed the squares as [tex]\(64 + 24\)[/tex] rather than [tex]\(64 + 144\)[/tex].
3. Omyra had an incorrect final sum of [tex]\(d^2 = 88\)[/tex] instead of the correct [tex]\(d^2 = 208\)[/tex].
Therefore, the detailed mistake identifications:
1. "Error in addition [tex]\( 64 + 24 \)[/tex], correct is [tex]\( 64 + 144 \)[/tex]."
2. "Correct calculation of [tex]\( d^2 \)[/tex] should be [tex]\( 144 + 64 = 208 \)[/tex]."
3. "Correct distance calculation [tex]\( d = \sqrt{208} \approx 14.42 \)[/tex]."
Omyra made errors in both the intermediate steps and the final distance calculation.
1. Determine the distance Joseph and Isabelle traveled:
- Joseph jogged north at a speed of 8 km/h for 1.5 hours.
[tex]\[ \text{Distance traveled by Joseph} = 8 \text{ km/h} \times 1.5 \text{ hours} = 12 \text{ km} \][/tex]
- Isabelle rode west at a speed of 12 km/h for 1.5 hours.
[tex]\[ \text{Distance traveled by Isabelle} = 12 \text{ km/h} \times 1.5 \text{ hours} = 18 \text{ km} \][/tex]
2. Use the Pythagorean theorem to find the distance between Joseph and Isabelle:
Since Joseph and Isabelle are moving perpendicular to each other (one north, one west), we apply the Pythagorean theorem:
[tex]\[ d^2 = (\text{Distance of Joseph})^2 + (\text{Distance of Isabelle})^2 \][/tex]
[tex]\[ d^2 = 12^2 + 18^2 \][/tex]
3. Calculate the squares:
[tex]\[ 12^2 = 144 \][/tex]
[tex]\[ 18^2 = 324 \][/tex]
4. Sum the squares:
[tex]\[ 144 + 324 = 468 \][/tex]
5. Calculate the square root to find the distance [tex]\(d\)[/tex]:
[tex]\[ d = \sqrt{468} \approx 21.63 \text{ km} \][/tex]
With these steps, we found the distance between Joseph and Isabelle to be approximately 21.63 km. Now let's identify Omyra's errors based on her work and the correct calculations above.
Omyra's work:
[tex]\[ 8^2 + 12^2 = d^2 \][/tex]
[tex]\[ 64 + 24 = d^2 \][/tex]
[tex]\[ 88 = d^2 \][/tex]
Errors in Omyra's calculations:
1. Error in squaring the speeds:
Omyra correctly squared the speeds initially in her work as follows:
- [tex]\(8^2 = 64\)[/tex]
- However, she incorrectly noted [tex]\(12^2 = 24\)[/tex] whereas it should be [tex]\(12^2 = 144\)[/tex].
2. Error in summing the squares:
The correctly squared values of distances should be:
[tex]\[ 64 + 144 = 208 \][/tex]
But Omyra incorrectly added:
[tex]\[ 64 + 24 = 88 \][/tex]
3. Error in calculating the final distance:
Omyra incorrectly used the sum [tex]\(d^2 = 88\)[/tex], leading to an incorrect final distance calculation. The correct step should be finding the square root of the correct sum:
[tex]\[ d = \sqrt{208} \approx 14.42 \text{ km} \][/tex]
Correct statements describing her errors:
1. Omyra incorrectly calculated [tex]\(12^2 \)[/tex] as [tex]\(24\)[/tex] instead of [tex]\(144\)[/tex].
2. Omyra incorrectly summed the squares as [tex]\(64 + 24\)[/tex] rather than [tex]\(64 + 144\)[/tex].
3. Omyra had an incorrect final sum of [tex]\(d^2 = 88\)[/tex] instead of the correct [tex]\(d^2 = 208\)[/tex].
Therefore, the detailed mistake identifications:
1. "Error in addition [tex]\( 64 + 24 \)[/tex], correct is [tex]\( 64 + 144 \)[/tex]."
2. "Correct calculation of [tex]\( d^2 \)[/tex] should be [tex]\( 144 + 64 = 208 \)[/tex]."
3. "Correct distance calculation [tex]\( d = \sqrt{208} \approx 14.42 \)[/tex]."
Omyra made errors in both the intermediate steps and the final distance calculation.
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