Let's analyze the situation step by step:
1. Determine the distances traveled by Joseph and Isabelle after 1.5 hours:
- Joseph jogged north at 8 kilometers per hour.
[tex]\[
\text{Distance traveled by Joseph} = 8 \text{ km/h} \times 1.5 \text{ h} = 12 \text{ km}
\][/tex]
- Isabelle rode her bike west at 12 kilometers per hour.
[tex]\[
\text{Distance traveled by Isabelle} = 12 \text{ km/h} \times 1.5 \text{ h} = 18 \text{ km}
\][/tex]
2. Calculate the squared distances:
- The square of the distance traveled by Joseph:
[tex]\[
12^2 = 144
\][/tex]
- The square of the distance traveled by Isabelle:
[tex]\[
18^2 = 324
\][/tex]
3. Sum the squares of the distances:
[tex]\[
144 + 324 = 468
\][/tex]
4. Using the Pythagorean theorem, find the distance [tex]\( d \)[/tex] between Joseph and Isabelle:
[tex]\[
d = \sqrt{468} \approx 21.63 \text{ km}
\][/tex]
Now, let's check Omyra's calculations and identify her errors:
- Omyra's first step is correct:
[tex]\[
8^2 + 12^2 = d^2
\][/tex]
- However, her calculation of the squares of the speeds was incorrect:
- [tex]\(8^2 = 64\)[/tex] is correct.
- [tex]\(12^2 = 144\)[/tex] is also correct, but she incorrectly added 24 instead of 144.
- Omyra incorrectly summed the squares as:
[tex]\[
64 + 24 = 88
\][/tex]
Instead, it should be:
[tex]\[
64 + 144 = 208
\][/tex]
- Since the square additions were incorrect, her conclusion [tex]\(d^2 = 88\)[/tex] is incorrect.
The correct statements describing her errors are:
- She calculated [tex]\(64 + 24\)[/tex] (which mistakenly assumes [tex]\(12^2 = 24\)[/tex]) instead of [tex]\(64 + 144\)[/tex].
- She concluded [tex]\(88 = d^2\)[/tex] instead of [tex]\(208 = d^2\)[/tex].
To summarize:
1. Omyra correctly identified [tex]\(8^2 + 12^2 = d^2\)[/tex].
2. She miscalculated [tex]\(12^2\)[/tex] leading to adding [tex]\(64 + 24\)[/tex] instead of [tex]\(64 + 144\)[/tex].
3. Her conclusion of [tex]\(88 = d^2\)[/tex] was based on incorrect arithmetic.
Her errors were in the calculation and summation of the squared distances.
