These questions are based on the information given below. A 72 km tunnel connects two cities P and Q. There are two gutters in the tunnel. Gutter 1 is the nearest gutter to P. The distance between gutter 1 and P is `16(2)/(3)%` less than the distance between gutter 2 and Q.
One day, a hospital in P received information that an accident had occurred at gutter 2. An ambulance started from P at 60 km/hr and 10 minutes later it crossed gutter 1. It doubled its speed after that travelled to gutter 2 and returned back at that speed. If it takes a total of 2 minutes to take the patient into and out of the ambulance, what is the total time taken by the ambulance to shift the patient to the hospital?

Options

A. 65 minutes
B. 67 minutes
C. 68 minutes
D. 69 minutes

Answer

To solve the problem step by step, we will break down the information provided and calculate the total time taken by the ambulance to shift the patient to the hospital.

### Step 1: Understand the distances involved
1. The total distance between cities P and Q is 72 km.
2. Let the distance between gutter 2 and Q be \( x \).
3. The distance between gutter 1 and P is \( 16\frac{2}{3}\% \) less than \( x \). This can be expressed as:
\[ \text{Distance from P to Gutter 1} = x - \left( \frac{16\frac{2}{3}}{100} \times x \right) = x - \frac{50}{3} \times \frac{x}{100} = \frac{5x}{6} \]

### Step 2: Set up the equation for total distance
4. The total distance from P to Gutter 1 and Gutter 2 can be expressed as:
\[ \frac{5x}{6} + x = 72 \]
5. To solve for \( x \):
\[ \frac{5x + 6x}{6} = 72 \implies \frac{11x}{6} = 72 \implies 11x = 72 \times 6 \implies 11x = 432 \implies x = \frac{432}{11} \approx 39.27 \text{ km} \]

### Step 3: Calculate the distance from P to Gutter 1
6. Now, substitute \( x \) back to find the distance from P to Gutter 1:
\[ \text{Distance from P to Gutter 1} = \frac{5x}{6} = \frac{5 \times \frac{432}{11}}{6} = \frac{2160}{66} = \frac{360}{11} \approx 32.73 \text{ km} \]

### Step 4: Calculate the distance from Gutter 1 to Gutter 2
7. The distance from Gutter 1 to Gutter 2 is:
\[ \text{Distance from Gutter 1 to Gutter 2} = 72 - \frac{360}{11} = \frac{792 - 360}{11} = \frac{432}{11} \approx 39.27 \text{ km} \]

### Step 5: Calculate the time taken by the ambulance
8. The ambulance travels from P to Gutter 1 at a speed of 60 km/hr. The time taken to reach Gutter 1 is:
\[ \text{Time to Gutter 1} = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{360}{11}}{60} = \frac{360}{660} \text{ hr} = \frac{6}{11} \text{ hr} \approx 32.73 \text{ minutes} \]

9. After 10 minutes, the ambulance crosses Gutter 1, so the time taken to reach Gutter 1 is:
\[ \text{Total time to Gutter 1} = 10 + \frac{6}{11} \text{ minutes} \approx 10 + 32.73 \approx 42.73 \text{ minutes} \]

### Step 6: Calculate the time to Gutter 2 and back
10. After crossing Gutter 1, the ambulance doubles its speed to 120 km/hr to reach Gutter 2. The time taken to reach Gutter 2 is:
\[ \text{Time to Gutter 2} = \frac{\frac{432}{11}}{120} = \frac{432}{1320} \text{ hr} = \frac{36}{110} \text{ hr} \approx 19.82 \text{ minutes} \]

11. The time taken to return from Gutter 2 to P at the same speed is also:
\[ \text{Time back to P} = 19.82 \text{ minutes} \]

### Step 7: Add the time for patient handling
12. The total time taken for handling the patient is 2 minutes. Therefore, the total time taken by the ambulance is:
\[ \text{Total time} = \text{Time to Gutter 1} + \text{Time to Gutter 2} + \text{Time back to P} + \text{Handling time} \]
\[ = 42.73 + 19.82 + 19.82 + 2 \approx 84.37 \text{ minutes} \]

### Final Answer
The total time taken by the ambulance to shift the patient to the hospital is approximately **84.37 minutes**.

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