Answer:
First, let's find the width of the rectangle field using the formula for the area of a rectangle:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given that the area is \( 3584 \, \text{m}^2 \) and the length is \( 64 \, \text{m} \), we can rearrange the formula to solve for the width:
\[ \text{Width} = \frac{\text{Area}}{\text{Length}} \]
\[ \text{Width} = \frac{3584 \, \text{m}^2}{64 \, \text{m}} \]
\[ \text{Width} = 56 \, \text{m} \]
Now, we can calculate the total distance around the field for one lap:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
\[ \text{Perimeter} = 2 \times (64 \, \text{m} + 56 \, \text{m}) \]
\[ \text{Perimeter} = 2 \times 120 \, \text{m} \]
\[ \text{Perimeter} = 240 \, \text{m} \]
To find out how long it will take to go around the field five times, we'll calculate the total distance he needs to cover:
\[ \text{Total distance} = 5 \times \text{Perimeter} \]
\[ \text{Total distance} = 5 \times 240 \, \text{m} \]
\[ \text{Total distance} = 1200 \, \text{m} \]
Now, we can convert his speed from kilometers per hour (km/h) to meters per hour (m/h) because the distance is in meters:
\[ 8 \, \text{km/h} = 8000 \, \text{m/h} \]
Now, we can calculate the time it will take:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
\[ \text{Time} = \frac{1200 \, \text{m}}{8000 \, \text{m/h}} \]
\[ \text{Time} = 0.15 \, \text{hours} \]
Finally, we can convert the time from hours to minutes:
\[ 0.15 \, \text{hours} \times 60 \, \text{minutes/hour} = 9 \, \text{minutes} \]
So, it will take him 9 minutes to go around the field five times.
