Given the problem, we need to determine how many hours it will take to travel a distance of 105 km without any halt using speed $S_5$.

Step 1: Define the speeds

Let's denote the speeds as follows:

• $S_1 = x$
• $S_2$
• $S_3$
• $S_4$
• $S_5 = x^2 + 15$

The relationships between the speeds are:

1. $S_5 = x^2 + 15$
2. $S_3 = \frac{S_5}{2} = \frac{x^2 + 15}{2}$
3. $S_2 = \frac{S_3}{2} = \frac{x^2 + 15}{4}$
4. $S_4 = 2(S_1 + S_2) = 2\left(x + \frac{x^2 + 15}{4}\right)$

Step 2: Calculate the total distance covered

The car travels at 5 different speeds for one hour each, and takes 4 halts. Therefore, the total time traveling at these speeds is 5 hours. The total distance is 105 km.

The distance covered with each speed is: $\text{Total Distance} = S_1 + S_2 + S_3 + S_4 + S_5 = 105 \text{ km}$

So: $x + \frac{x^2 + 15}{4} + \frac{x^2 + 15}{2} + 2\left(x + \frac{x^2 + 15}{4}\right) + (x^2 + 15) = 105$

Step 3: Simplify the equation

Let's simplify each term: $x + \frac{x^2 + 15}{4} + \frac{x^2 + 15}{2} + 2\left(x + \frac{x^2 + 15}{4}\right) + x^2 + 15 = 105$

Simplify the terms within the equation:

• Combine $\frac{x^2 + 15}{4} + 2\left(x + \frac{x^2 + 15}{4}\right)$ into one expression.
• Combine all the x's and the $x^2$ terms.

But, let's directly calculate time for $S_5$ as the problem asks for it:

• Since $S_5 = x^2 + 15$ and the total distance is 105 km.

Step 4: Calculate time taken at speed $S_5$

The time taken to travel 105 km at speed $S_5$ is: $\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{105}{S_5} = \frac{105}{x^2 + 15}$

Step 5: Solve for time using choices

Substitute each option and see if it satisfies the original equation.

• Option A: 4 hours $4 = \frac{105}{x^2 + 15} \Rightarrow x^2 + 15 = \frac{105}{4} = 26.25 \Rightarrow x^2 = 11.25$

• Option B: 105/30 hours $\frac{105}{30} = \frac{105}{x^2 + 15} \Rightarrow x^2 + 15 = 30 \Rightarrow x^2 = 15$

• Option C: 105/40 hours $\frac{105}{40} = \frac{105}{x^2 + 15} \Rightarrow x^2 + 15 = 40 \Rightarrow x^2 = 25$

• Option D: 105/35 hours $\frac{105}{35} = \frac{105}{x^2 + 15} \Rightarrow x^2 + 15 = 35 \Rightarrow x^2 = 20$

Comparing the simplest time to match options: $x^2 = 25$, $x = 5$.

Correct answer: $105/40$ (Option C)

So, the time taken to travel the distance without any halt at speed $S_5$ is $\boxed{105/40}$ or 2.625 hours.

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Do you want details or have any questions? Here are some related questions:

1. How does time vary with speed when traveling the same distance?
2. How can we simplify the calculation process in problems involving multiple variables?
3. What is the relationship between distance, speed, and time?
4. What other types of problems can be solved with the concept of speed and time?
5. How do real-world constraints, like halts and speed changes, affect travel time?

Tip: Breaking down complex equations into simpler parts can make problem-solving easier and more efficient.