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Sagot :
Let's find out where Omyra went wrong in her calculations step-by-step.
1. Speeds and Time:
- Joseph jogs north at a speed of 8 kilometers per hour.
- Isabelle rides west at a speed of 12 kilometers per hour.
- The time elapsed is 1.5 hours.
2. Distances Traveled:
- Distance traveled by Joseph: [tex]\(8 \text{ km/h} \times 1.5 \text{ hours} = 12 \text{ kilometers}\)[/tex].
- Distance traveled by Isabelle: [tex]\(12 \text{ km/h} \times 1.5 \text{ hours} = 18 \text{ kilometers}\)[/tex].
3. Distance Apart (Using the Pythagorean Theorem):
- Let [tex]\( a \)[/tex] be the distance between Joseph and Isabelle after 1.5 hours.
- According to the Pythagorean theorem:
[tex]\[ a^2 = (\text{distance traveled by Joseph})^2 + (\text{distance traveled by Isabelle})^2 \][/tex]
- Therefore:
[tex]\[ a^2 = 12^2 + 18^2 \][/tex]
- Calculating the squares:
[tex]\[ 12^2 = 144 \][/tex]
[tex]\[ 18^2 = 324 \][/tex]
- Adding these values:
[tex]\[ a^2 = 144 + 324 = 468 \][/tex]
- Taking the square root to find [tex]\( a \)[/tex]:
[tex]\[ a = \sqrt{468} \approx 21.633 \][/tex]
4. Omyra's Errors:
- Omyra's initial steps were correct when she set up the equation using the Pythagorean theorem: [tex]\(8^2 + 12^2 = a^2\)[/tex].
- However, she made two errors in her calculations:
- She incorrectly calculated the square of 12 as 24. The correct value should be [tex]\( 12^2 = 144 \)[/tex].
- Her resulting equation was: [tex]\( 64 + 24 = a^2 \)[/tex], which summed to 88 instead of the correct [tex]\( 64 + 144 = 208 \)[/tex].
- This led to a wrong intermediate step.
Correcting Omyra’s errors, Joseph and Isabelle were actually [tex]\( \approx 21.633 \)[/tex] kilometers apart after 1.5 hours.
Errors in Omyra’s Work:
- She incorrectly calculated [tex]\( 12^2 \)[/tex] as 24 instead of 144.
- Her sum was [tex]\( 64 + 24 = 88 \)[/tex] instead of [tex]\( 64 + 144 = 208 \)[/tex].
Therefore, the statements that describe Omyra's errors are:
- Omyra incorrectly calculated [tex]\( 12^2 \)[/tex].
- Omyra incorrectly added the squares of the distances.
These mistakes led to an incorrect final distance between Joseph and Isabelle.
1. Speeds and Time:
- Joseph jogs north at a speed of 8 kilometers per hour.
- Isabelle rides west at a speed of 12 kilometers per hour.
- The time elapsed is 1.5 hours.
2. Distances Traveled:
- Distance traveled by Joseph: [tex]\(8 \text{ km/h} \times 1.5 \text{ hours} = 12 \text{ kilometers}\)[/tex].
- Distance traveled by Isabelle: [tex]\(12 \text{ km/h} \times 1.5 \text{ hours} = 18 \text{ kilometers}\)[/tex].
3. Distance Apart (Using the Pythagorean Theorem):
- Let [tex]\( a \)[/tex] be the distance between Joseph and Isabelle after 1.5 hours.
- According to the Pythagorean theorem:
[tex]\[ a^2 = (\text{distance traveled by Joseph})^2 + (\text{distance traveled by Isabelle})^2 \][/tex]
- Therefore:
[tex]\[ a^2 = 12^2 + 18^2 \][/tex]
- Calculating the squares:
[tex]\[ 12^2 = 144 \][/tex]
[tex]\[ 18^2 = 324 \][/tex]
- Adding these values:
[tex]\[ a^2 = 144 + 324 = 468 \][/tex]
- Taking the square root to find [tex]\( a \)[/tex]:
[tex]\[ a = \sqrt{468} \approx 21.633 \][/tex]
4. Omyra's Errors:
- Omyra's initial steps were correct when she set up the equation using the Pythagorean theorem: [tex]\(8^2 + 12^2 = a^2\)[/tex].
- However, she made two errors in her calculations:
- She incorrectly calculated the square of 12 as 24. The correct value should be [tex]\( 12^2 = 144 \)[/tex].
- Her resulting equation was: [tex]\( 64 + 24 = a^2 \)[/tex], which summed to 88 instead of the correct [tex]\( 64 + 144 = 208 \)[/tex].
- This led to a wrong intermediate step.
Correcting Omyra’s errors, Joseph and Isabelle were actually [tex]\( \approx 21.633 \)[/tex] kilometers apart after 1.5 hours.
Errors in Omyra’s Work:
- She incorrectly calculated [tex]\( 12^2 \)[/tex] as 24 instead of 144.
- Her sum was [tex]\( 64 + 24 = 88 \)[/tex] instead of [tex]\( 64 + 144 = 208 \)[/tex].
Therefore, the statements that describe Omyra's errors are:
- Omyra incorrectly calculated [tex]\( 12^2 \)[/tex].
- Omyra incorrectly added the squares of the distances.
These mistakes led to an incorrect final distance between Joseph and Isabelle.
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