Calculating Walk Time Around a Square Field

This problem asks us to find the time a man takes to walk around the boundary of a square field given its area and his walking speed. To solve this, we need to determine the total distance the man walks (the perimeter of the square field) and then use the formula: Time = Distance / Speed.

Step 1: Find the Area in Square Meters

The area of the square field is given as 25 hectares.

We know that 1 hectare is equal to 10,000 square meters.

So, the area in square meters is:

\(\text{Area} = 25 \text{ hectares} \times 10,000 \frac{\text{m}^2}{\text{hectare}}\)

\(\text{Area} = 250,000 \text{ m}^2\)

Step 2: Calculate the Side Length of the Square Field

The area of a square is calculated by the formula:

\(\text{Area} = (\text{Side})^2\)

To find the side length, we take the square root of the area:

\(\text{Side} = \sqrt{\text{Area}}\)

\(\text{Side} = \sqrt{250,000 \text{ m}^2}\)

\(\text{Side} = \sqrt{25 \times 10,000} \text{ m}\)

\(\text{Side} = \sqrt{25} \times \sqrt{10,000} \text{ m}\)

\(\text{Side} = 5 \times 100 \text{ m}\)

\(\text{Side} = 500 \text{ m}\)

The side length of the square field is 500 meters.

Step 3: Determine the Distance (Perimeter of the Square)

The man walks around the boundary, which means he walks the perimeter of the square field.

The perimeter of a square is calculated by the formula:

\(\text{Perimeter} = 4 \times \text{Side}\)

\(\text{Perimeter} = 4 \times 500 \text{ m}\)

\(\text{Perimeter} = 2000 \text{ m}\)

The total distance the man walks is 2000 meters.

Step 4: Convert Speed to Consistent Units

The man's speed is given as 5 km/hr. We need to convert this speed to meters per minute so that the distance (in meters) and speed units are consistent for calculating time in minutes.

• Convert kilometers to meters: 1 km = 1000 m
• Convert hours to minutes: 1 hour = 60 minutes

So, the speed in meters per minute is:

\(\text{Speed} = 5 \frac{\text{km}}{\text{hr}} = 5 \times \frac{1000 \text{ m}}{60 \text{ minutes}}\)

\(\text{Speed} = \frac{5000}{60} \frac{\text{m}}{\text{minute}}\)

\(\text{Speed} = \frac{500}{6} \frac{\text{m}}{\text{minute}}\)

\(\text{Speed} = \frac{250}{3} \frac{\text{m}}{\text{minute}}\)

The man's speed is \(\frac{250}{3}\) meters per minute.

Step 5: Calculate the Time Taken

Now we use the formula: Time = Distance / Speed

\(\text{Time} = \frac{2000 \text{ m}}{\frac{250}{3} \frac{\text{m}}{\text{minute}}}\)

\(\text{Time} = 2000 \times \frac{3}{250} \text{ minutes}\)

\(\text{Time} = \frac{2000 \times 3}{250} \text{ minutes}\)

\(\text{Time} = \frac{6000}{250} \text{ minutes}\)

Simplify the fraction:

\(\text{Time} = \frac{600}{25} \text{ minutes}\)

\(\text{Time} = 24 \text{ minutes}\)

The man will take 24 minutes to walk around the boundary of the square field.

Revision Table: Formulas and Concepts

Additional Information: Units and Conversions

Understanding units and how to convert them is crucial in physics and word problems involving measurements. Always ensure all quantities are in compatible units before performing calculations. For example, if distance is in meters, speed should be in meters per unit of time (like seconds or minutes), and the resulting time will be in that same unit.

• Distance Units: meters (m), kilometers (km), centimeters (cm), miles, feet, etc.
• Area Units: square meters (m\(^2\)), square kilometers (km\(^2\)), hectares (ha), acres, etc.
• Speed Units: meters per second (m/s), kilometers per hour (km/hr), miles per hour (mph), meters per minute (m/min), etc.
• Time Units: seconds (s), minutes (min), hours (hr), days, etc.

Common conversions:

• 1 km = 1000 m
• 1 hour = 60 minutes
• 1 minute = 60 seconds
• 1 hour = 3600 seconds
• 1 hectare = 10,000 m\(^2\)

When solving problems, choose a consistent set of units for distance and time based on what is given or what is asked for in the final answer.