The problem describes a drawing of a swimming pool, where:

1. The current drawing scale is $1$ inch $= 3$ meters.
2. The drawing shows a pool with a length of $12$ inches and a width of $5$ inches.

The task is to find the length of this pool drawing using a new scale of $2$ inches $= 5$ meters.

Step-by-Step Solution:

1. Determine the Actual Pool Length and Width:

   • With the current scale of $1$ inch $= 3$ meters, a $12$-inch length on paper represents: $12 \text{ inches} \times 3 \text{ meters/inch} = 36 \text{ meters}$
   • Similarly, the $5$-inch width represents: $5 \text{ inches} \times 3 \text{ meters/inch} = 15 \text{ meters}$

   So, the actual dimensions of the pool are $36$ meters in length and $15$ meters in width.

2. Convert Using the New Scale:

   • The new scale is $2$ inches $= 5$ meters. To find the new drawing dimensions:
     • For the length: $\frac{36 \text{ meters}}{5 \text{ meters}} \times 2 \text{ inches} = 14.4 \text{ inches}$
     • For the width: $\frac{15 \text{ meters}}{5 \text{ meters}} \times 2 \text{ inches} = 6 \text{ inches}$

3. Answer:

   • The length of the drawing with the new scale will be $14.4$ inches.

Explanation of Scale Utility

Some scales might be more useful depending on available drawing space, clarity, and purpose. The new scale (2 inches = 5 meters) is larger, which can be more detailed and easier to interpret but takes up more space.

Would you like further explanation or have any questions?

Additional Questions to Explore:

1. How would the width of the pool drawing change under a different scale?
2. How can you determine the real-life dimensions from any given scale and drawing dimensions?
3. Why might smaller scales (e.g., 1 inch = 10 meters) be beneficial in architectural plans?
4. If the scale was 1 cm = 2 meters, what would the drawing’s dimensions be?
5. How can you verify the scale of a drawing when only given dimensions?

Tip:

When working with scale conversions, remember to keep track of units carefully to avoid errors, especially when switching between inches and meters.