Let's solve this step-by-step.
Let's denote:
- [tex]\( x \)[/tex] as Lindsey's distance from the campsite at 2:00 p.m.
- Kara's distance at 2:00 p.m., since it's twice Lindsey's distance, is [tex]\( 2x \)[/tex].
By 2:30 p.m.:
- Lindsey has added 0.5 miles to her distance, so her total distance is [tex]\( x + 0.5 \)[/tex].
- Kara has added 1 mile to her distance, so her total distance is [tex]\( 2x + 1 \)[/tex].
Now we consider the triangle formed by the points:
- The campsite (the origin, or [tex]\((0,0)\)[/tex]),
- Lindsey's position [tex]\((0, x + 0.5)\)[/tex],
- Kara's position [tex]\((2x + 1, 0)\)[/tex].
We need the area of this right-angled triangle:
- The base of the triangle is along the x-axis, which is Kara's distance at 2:30 p.m.: [tex]\( 2x + 1 \)[/tex] miles.
- The height of the triangle is along the y-axis, which is Lindsey's distance at 2:30 p.m.: [tex]\( x + 0.5 \)[/tex] miles.
The area [tex]\( A \)[/tex] of a right-angled triangle is given by:
[tex]\[ A = \frac{1}{2} \times (\text{base}) \times (\text{height}) \][/tex]
So we have:
[tex]\[ A = \frac{1}{2} \times (2x + 1) \times (x + 0.5) \][/tex]
Now, we simplify:
[tex]\[ A = \frac{1}{2} \times (2x + 1) \times (x + 0.5) \][/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]
[tex]\[ A = x^2 + x + 0.25 \][/tex]
Thus, the function that represents the area of the triangular region formed by the girls' locations and the campsite at 2:30 p.m. is:
[tex]\[ f(x) = x^2 + x + 0.25 \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{B. \, f(x) = x^2 + x + 0.25} \][/tex]
