Question

1. due to all the afternoon thunderstorms in

south florida a new covered shelter is to be
built on the recreational trail. using the ideas
from partitioning a line segment to find the
location of the covered shelter.
the directed distance from the trail entrance
to the new covered shelter to the opposite
park boundary should be a ratio of 2:3.
in other words, trail entrance to the covered
shelter is 2 parts and then the distance from
the covered shelter to the opposite park
boundary is 3 parts. assume the trail entrance is at a
midpoint
20(a)show your work neatly in the blank space
20(b) graph the covered shelter on the map
include a symbol on the map key

Answer:

To solve this problem, we assume the trail entrance is at the midpoint of the recreational trail. Let's denote the length of the trail as \( L \). Since the trail entrance is at the midpoint, the distance from the trail entrance to either end (opposite park boundary) is \( \frac{L}{2} \).

Step 1: Define the ratio

The ratio of the distance from the trail entrance to the covered shelter (\( d_1 \)) to the distance from the covered shelter to the opposite park boundary (\( d_2 \)) is \( 2:3 \). So, \( \frac{d_1}{d_2} = \frac{2}{3} \), which implies \( d_2 = \frac{3}{2}d_1 \).

Step 2: Relate to the total distance

We know that \( d_1 + d_2 = \frac{L}{2} \) (since the trail entrance is at the midpoint). Substituting \( d_2 = \frac{3}{2}d_1 \) into this equation:
\[
d_1 + \frac{3}{2}d_1 = \frac{L}{2}
\]
\[
\frac{2d_1 + 3d_1}{2} = \frac{L}{2}
\]
\[
\frac{5d_1}{2} = \frac{L}{2}
\]
Multiplying both sides by 2:
\[
5d_1 = L
\]
\[
d_1 = \frac{L}{5}
\]

Step 3: Determine the location

Since the trail entrance is at the midpoint (\( \frac{L}{2} \) from either end), the distance from the trail entrance to the covered shelter is \( \frac{L}{5} \). So, the location of the covered shelter from the start of the trail (let's assume the start is at 0) would be:

• If the trail entrance is at \( \frac{L}{2} \), and we move \( \frac{L}{5} \) towards the opposite park boundary, the position is \( \frac{L}{2} + \frac{L}{5} = \frac{5L + 2L}{10} = \frac{7L}{10} \) from the start.
• Alternatively, from the opposite park boundary (which is at \( L \)), the distance is \( L - \frac{7L}{10} = \frac{3L}{10} \), which matches the ratio \( 2:3 \) (since \( \frac{7L}{10} - \frac{L}{2} = \frac{7L - 5L}{10} = \frac{2L}{10} = \frac{L}{5} \) and \( L - \frac{7L}{10} = \frac{3L}{10} \), so \( \frac{L}{5} : \frac{3L}{10} = 2:3 \)).

Step 4: Graphing (20b)

1. Draw a line segment representing the recreational trail.
2. Mark the midpoint as the trail entrance (e.g., with a triangle).
3. Divide the segment from the trail entrance to the opposite park boundary into 3 equal parts (since the ratio is 2:3, the total parts from entrance to boundary is 5? Wait, no—wait, the ratio is entrance to shelter (2 parts) and shelter to boundary (3 parts). So from entrance to boundary, there are 2 + 3 = 5 parts? Wait, no, the entrance is at the midpoint, so the distance from entrance to boundary is \( \frac{L}{2} \). So we need to divide \( \frac{L}{2} \) into 5 parts? Wait, no, the ratio is 2:3, so the total number of parts is 2 + 3 = 5. So each part is \( \frac{\frac{L}{2}}{5} = \frac{L}{10} \). Therefore, the distance from entrance to shelter is \( 2 \times \frac{L}{10} = \frac{L}{5} \), and from shelter to boundary is \( 3 \times \frac{L}{10} = \frac{3L}{10} \), which matches our earlier calculation.

So, to graph:

• Draw the trail as a line.
• Mark the start (0), end (L), and midpoint (trail entrance, \( \frac{L}{2} \)).
• From the trail entrance, move \( \frac{L}{5} \) towards the end (opposite park boundary) and mark that point as the covered shelter (e.g., with a square).
• Include a key: triangle for trail entrance, square for covered shelter.

Final Answer (for 20a)

The location of the covered shelter is \( \frac{7L}{10} \) from the start of the trail (or \( \frac{3L}{10} \) from the opposite park boundary), determined by partitioning the segment from the trail entrance to the boundary in the ratio 2:3.

For 20b, the graph should show the trail, trail entrance (midpoint), and covered shelter located \( \frac{L}{5} \) from the trail entrance towards the opposite park boundary, with appropriate symbols in the key.