A woman starts from her home at `9.00` a. m., walks with a speed of `5 km h^(-1)` on straight road up to her office `2.5 km`away, stays at the offiec up to `5.00 p. m.`, and returnshome by anauto with a seed of `25 kmh^(-1)`. Plot the position-time graph of the woman taking home as origin.

Explanation

To solve the problem step by step, we will analyze the woman's journey from her home to her office and back, and then plot the position-time graph based on the information provided.

### Step 1: Calculate the time taken to walk to the office
The woman walks from her home to her office, which is 2.5 km away, at a speed of 5 km/h.

**Formula:**
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

**Calculation:**
\[ \text{Time} = \frac{2.5 \text{ km}}{5 \text{ km/h}} = 0.5 \text{ hours} = 30 \text{ minutes} \]

### Step 2: Determine the time of arrival at the office
She starts her journey at 9:00 a.m. and takes 30 minutes to reach the office.

**Calculation:**
\[ 9:00 \text{ a.m.} + 30 \text{ minutes} = 9:30 \text{ a.m.} \]

### Step 3: Determine the duration of stay at the office
The woman stays at the office until 5:00 p.m.

### Step 4: Calculate the time taken to return home by auto
The woman returns home by auto at a speed of 25 km/h. The distance is still 2.5 km.

**Formula:**
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

**Calculation:**
\[ \text{Time} = \frac{2.5 \text{ km}}{25 \text{ km/h}} = 0.1 \text{ hours} = 6 \text{ minutes} \]

### Step 5: Determine the time of arrival back home
She leaves the office at 5:00 p.m. and takes 6 minutes to return home.

**Calculation:**
\[ 5:00 \text{ p.m.} + 6 \text{ minutes} = 5:06 \text{ p.m.} \]

### Step 6: Plot the position-time graph
1. **From 9:00 a.m. to 9:30 a.m.**: The woman moves from home (0 km) to office (2.5 km). This will be a straight line going up to 2.5 km.
2. **From 9:30 a.m. to 5:00 p.m.**: The woman stays at the office, so the position remains constant at 2.5 km. This will be a horizontal line at 2.5 km.
3. **From 5:00 p.m. to 5:06 p.m.**: The woman returns home from the office. This will be a straight line going down from 2.5 km back to 0 km.

### Summary of the position-time graph:
- **0 km at 9:00 a.m.**
- **2.5 km at 9:30 a.m.**
- **2.5 km from 9:30 a.m. to 5:00 p.m.**
- **0 km at 5:06 p.m.**