We are tasked with determining the depth $d$ of a well, given that the total time between dropping a stone and hearing the splash is 5 seconds. The total time is the sum of two components:

• $t_1$ is the time it takes for the stone to fall down the well.
• $t_2$ is the time it takes for the sound to travel back up the well.

We are provided with the following relationships:

1. The time for the stone to fall is $t_1 = \frac{d}{4}$.
2. The time for the sound to travel back is $t_2 = \frac{d}{1090}$.

The total time elapsed is the sum of these two times: $t_1 + t_2 = \frac{d}{4} + \frac{d}{1090}$ We are also given that the total time is 5 seconds, so: $\frac{d}{4} + \frac{d}{1090} = 5$

Step 1: Combine the terms with a common denominator

To combine the fractions, we find a common denominator. The least common denominator between 4 and 1090 is 4360. Rewriting both fractions:

$\frac{d}{4} = \frac{1090d}{4360}, \quad \frac{d}{1090} = \frac{4d}{4360}$

Now we can combine the fractions: $\frac{1090d + 4d}{4360} = 5$ $\frac{1094d}{4360} = 5$

Step 2: Solve for $d$

Now, multiply both sides of the equation by 4360 to eliminate the denominator: $1094d = 5 \times 4360$ $1094d = 21800$

Now, divide both sides by 1094: $d = \frac{21800}{1094} \approx 19.92$

Final Answer:

The depth of the well is approximately $d = 19.9$ feet.

Would you like more details or clarifications on any part of the solution?

Here are 5 related questions to further your understanding:

1. What assumptions are made in this problem about the effect of air resistance?
2. How would the solution change if the speed of sound was different, such as in warmer or colder air?
3. What formula would apply if the well were on a different planet with a different gravitational constant?
4. How would you solve this if the total time was not 5 seconds, but another value?
5. How does the speed of sound in water compare to that in air, and how would this affect the calculation if the splash occurred underwater?

Tip: Always pay attention to units, especially when mixing speeds (like sound) and distances (like depth), as they must be consistent to avoid errors!