Let's break down and analyze the problem step-by-step, identifying where Omyra made mistakes in her calculations:
1. Understanding the Movement of Joseph and Isabelle:
- Joseph jogs north at a speed of 8 kilometers per hour.
- Isabelle rides her bike west at a speed of 12 kilometers per hour.
- Both travel for 1.5 hours.
2. Calculating Distances Traveled:
- Joseph’s Distance:
[tex]\[
\text{Joseph's distance} = \text{speed} \times \text{time} = 8 \, \text{km/h} \times 1.5 \, \text{hours} = 12 \, \text{km}
\][/tex]
- Isabelle’s Distance:
[tex]\[
\text{Isabelle's distance} = \text{speed} \times \text{time} = 12 \, \text{km/h} \times 1.5 \, \text{hours} = 18 \, \text{km}
\][/tex]
3. Using the Pythagorean Theorem to Find the Distance Apart:
In this scenario, Joseph’s and Isabelle’s traveled paths form the two legs of a right triangle, with the distance apart being the hypotenuse [tex]\(d\)[/tex].
- According to the Pythagorean theorem:
[tex]\[
a^2 + b^2 = d^2
\][/tex]
where [tex]\(a = 12 \, \text{km}\)[/tex] and [tex]\(b = 18 \, \text{km}\)[/tex].
- Calculate:
[tex]\[
12^2 + 18^2 = d^2
\][/tex]
[tex]\[
144 + 324 = d^2
\][/tex]
[tex]\[
468 = d^2
\][/tex]
- Solving for [tex]\(d\)[/tex]:
[tex]\[
d = \sqrt{468} \approx 21.633 \, \text{km}
\][/tex]
4. Identifying Omyra's Mistakes:
In Omyra’s work:
- The equation she attempted to use was correctly set up initially:
[tex]\[
8^2 + 12^2 = d^2
\][/tex]
- However, she made the following errors:
- Substitution Error:
In step [tex]$64 + 24 = d^2$[/tex], the correct summation should be:
[tex]\[
8^2 + 12^2 = 64 + 144 = 208
\][/tex]
Therefore, the correct form should be:
[tex]\[
64 + 144 = d^2
\][/tex]
- Calculation Error:
The statement [tex]$88 = a^2$[/tex] is incorrect because [tex]$64 + 24$[/tex] was mistakenly used instead of [tex]$64 + 144$[/tex]. Instead, it should be [tex]$208 = d^2$[/tex].
Based on the above analysis, the statements describing Omyra's errors are:
- [tex]$64 + 24 = d^2$[/tex]
- [tex]$88 = a^2$[/tex]
These statements indicate the calculation errors she made in the process.
