Problem Breakdown:

The diagram shows a running track consisting of a rectangular area with semi-circular ends. You're tasked with calculating the total length of the inside and outside of the track.

• Dimensions of the inside track:
  • Straight part (length of the rectangle) = 90 meters
  • Radius of the semi-circles = 30 meters

For part a, we will calculate the total length of the inside of the track.

Part a) Total length of the inside of the track:

1. Length of the straight sections:
   The straight sections are simply the two sides of the rectangle.
   $\text{Total length of straight sections} = 2 \times 90 = 180 \text{ meters}$

2. Length of the semi-circular sections:
   The length of each semi-circle is half the circumference of a full circle.
   The formula for the circumference of a circle is $2\pi r$, and since each end is a semi-circle, we only need half of this: $\text{Length of one semi-circle} = \pi \times r = \pi \times 30$ $\text{Total length of both semi-circles} = 2 \times \pi \times 30 \approx 188.4 \text{ meters}$

3. Total length of the inside of the track:
   $\text{Total length} = 180 + 188.4 = 368.4 \text{ meters}$ Rounding to the nearest meter: $\boxed{368 \text{ meters}}$

Part b) Total length of the outside of the track:

Now, for the outside of the track:

• The length of the straight sections is 90 meters on the inside, but the outside is 10 meters farther (since 40m is the distance to the outer edge, compared to 30m for the inner edge).
  The new radius for the semi-circles is $40 \text{ meters}$.

1. Straight sections of the outside:
   $\text{Total length of straight sections (outside)} = 2 \times 90 = 180 \text{ meters}$

2. Semi-circles on the outside:
   $\text{Length of one semi-circle (outside)} = \pi \times 40 \approx 125.66 \text{ meters}$ $\text{Total length of both semi-circles (outside)} = 2 \times \pi \times 40 \approx 251.3 \text{ meters}$

3. Total length of the outside of the track:
   $\text{Total length (outside)} = 180 + 251.3 = 431.3 \text{ meters}$ Rounding to the nearest meter: $\boxed{431 \text{ meters}}$

Additional Distance a Runner on the Outside Would Travel:

Finally, we can determine how much further a runner would have to travel on the outside of the track compared to the inside: $\text{Extra distance} = 431 - 368 = 63 \text{ meters}$ Thus, the runner on the outside would travel 63 meters more than the runner on the inside.

Do you want further details or have any questions? Here are 5 related problems to explore:

1. What would happen if the radius for the inner semi-circles was changed to 35 meters?
2. How would you calculate the area enclosed by the running track?
3. What is the average speed of a runner if they complete 5 laps of the inner track in 7 minutes?
4. Can you calculate the width of the track based on the difference between the inner and outer radii?
5. How would the length of the track change if the straight section was extended by 10 meters?

Tip: Remember that the formula for the circumference of a circle is key for many geometric problems, especially those involving arcs!