Let's carefully work through the problem step-by-step.
1. Understanding the Distances Covered:
- At 2:00 p.m., let Lindsey's distance from the campsite be [tex]\( x \)[/tex] miles.
- Hence, Kara’s distance at 2:00 p.m. is twice that of Lindsey’s, which is [tex]\( 2x \)[/tex] miles.
2. Distances at 2:30 p.m.:
- By 2:30 p.m., Kara covered an additional 1 mile. Therefore, her total distance is [tex]\( 2x + 1 \)[/tex] miles.
- Lindsey covered an additional 0.5 miles. Therefore, her total distance is [tex]\( x + 0.5 \)[/tex] miles.
3. Geometry of the Problem:
- The triangle is formed with:
- One vertex at the campsite,
- One vertex at Kara’s location,
- One vertex at Lindsey’s location.
- Kara’s location is [tex]\( (2x + 1, 0) \)[/tex] and Lindsey’s location is [tex]\( (0, x + 0.5) \)[/tex].
4. Calculating the Area of the Triangle:
- The base of the triangle is the distance Lindsey covered: [tex]\( x + 0.5 \)[/tex] miles.
- The height of the triangle is the distance Kara covered: [tex]\( 2x + 1 \)[/tex] miles.
- The area [tex]\( A \)[/tex] of a triangle is given by the formula:
[tex]\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\][/tex]
- Plugging in the actual base and height values:
[tex]\[
A = \frac{1}{2} \times (x + 0.5) \times (2x + 1)
\][/tex]
5. Simplifying the Area Expression:
- Distribute to simplify the product inside the parenthesis:
[tex]\[
A = \frac{1}{2} \times (x \cdot 2x + x \cdot 1 + 0.5 \cdot 2x + 0.5 \cdot 1)
\][/tex]
[tex]\[
A = \frac{1}{2} \times (2x^2 + x + x + 0.5)
\][/tex]
[tex]\[
A = \frac{1}{2} \times (2x^2 + 2x + 0.5)
\][/tex]
- Continue to simplify:
[tex]\[
A = \frac{1}{2} \times 2x^2 + \frac{1}{2} \times 2x + \frac{1}{2} \times 0.5
\][/tex]
[tex]\[
A = x^2 + x + 0.25
\][/tex]
6. Conclusion:
- The function representing the area of the triangular region is:
[tex]\[
f(x) = x^2 + x + 0.25
\][/tex]
- Among the given options, this matches with:
[tex]\[
\text{C. } f(x) = x^2 + x + 0.25
\][/tex]
Thus, the correct answer is:
C. [tex]\( f(x) = x^2 + x + 0.25 \)[/tex]
