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Joseph and Isabelle left Omyra's house at the same time. Joseph jogged north at 8 kilometers per hour, while Isabelle rode her bike west at 12 kilometers per hour. Omyra tried to figure out how far apart they were after 1.5 hours. Her work is shown below. Which statements describe her errors? Check all that apply.

[tex]\[
\begin{aligned}
8^2 + 12^2 &= a^2 \\
64 + 144 &= a^2 \\
208 &= a^2
\end{aligned}
\][/tex]

A. Omyra used the correct formula but made an arithmetic error.

B. Omyra should have multiplied the speeds by 1.5 hours before squaring them.

C. Omyra correctly applied the Pythagorean theorem.

D. Omyra's calculation of [tex]\( a^2 \)[/tex] is correct, but her arithmetic is incorrect.



Answer :

GinnyAnswer
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.

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