To determine the parametric equations for the path of the cyclist, let's break down the problem step-by-step:
1. Identify Initial and Final Positions:
- The intersection is at the origin [tex]\((0,0)\)[/tex].
- The start of the bicycle path is 5 miles east of the intersection, so the starting point is [tex]\((5,0)\)[/tex].
- The end of the bicycle path is 8 miles north of the intersection, so the ending point is [tex]\((0,8)\)[/tex].
2. Determine the time required:
- The cyclist rides from the start to the end in 2 hours. Thus, the total time is 2 hours.
3. Derive Parametric Equations:
The parametric equations for a straight-line motion between two points typically take the form:
[tex]\[
x(t) = x_{\text{initial}} + (x_{\text{final}} - x_{\text{initial}}) \frac{t}{T}
\][/tex]
[tex]\[
y(t) = y_{\text{initial}} + (y_{\text{final}} - y_{\text{initial}}) \frac{t}{T}
\][/tex]
where:
- [tex]\(x_{\text{initial}} = 5\)[/tex]
- [tex]\(y_{\text{initial}} = 0\)[/tex]
- [tex]\(x_{\text{final}} = 0\)[/tex]
- [tex]\(y_{\text{final}} = 8\)[/tex]
- [tex]\(T = 2\)[/tex]
Using these values,
[tex]\[
x(t) = 5 + (0 - 5) \frac{t}{2}
\][/tex]
[tex]\[
y(t) = 0 + (8 - 0) \frac{t}{2}
\][/tex]
Which simplifies to,
[tex]\[
x(t) = 5 - \frac{5}{2} t
\][/tex]
[tex]\[
y(t) = 4 t
\][/tex]
Thus, the parametric equations that model the path of the cyclist are:
[tex]\[
x(t) = 5 - \frac{5}{2} t
\][/tex]
[tex]\[
y(t) = 4 t
\][/tex]
Therefore, the correct answer is:
[tex]\[
x(t)=5 - \frac{5}{2}t \quad \text{and} \quad y(t)=4t
\][/tex]
