Lock of a safe consists of five discs each of which features the digits 0, 1, 2, ….., 9. The safe can be opened by dialing a special combination of the digits. If the work day lasts 13 hours and to dial one combination of digits takes 5 seconds, then number of days sufficient enough to open the safe, are

Options

A. 9
B. 10
C. 11
D. 12

Answer

To solve the problem, we need to follow these steps:

### Step 1: Calculate the total number of combinations for the lock
Each of the 5 discs can display any digit from 0 to 9, giving us 10 options per disc. Therefore, the total number of combinations can be calculated as:

\[ \text{Total combinations} = 10^5 \]

### Step 2: Calculate the total time to try all combinations
Since it takes 5 seconds to dial one combination, the total time in seconds to try all combinations is:

\[ \text{Total time (seconds)} = \text{Total combinations} \times \text{Time per combination} = 10^5 \times 5 \]

### Step 3: Convert total time from seconds to hours
To convert seconds to hours, we divide the total time in seconds by the number of seconds in an hour (3600 seconds):

\[ \text{Total time (hours)} = \frac{10^5 \times 5}{3600} \]

### Step 4: Calculate the total working hours in a day
The problem states that the workday lasts 13 hours. Therefore, we need to find out how many days it will take to try all combinations by dividing the total hours by the working hours per day:

\[ \text{Total days} = \frac{\text{Total time (hours)}}{13} \]

### Step 5: Calculate the final answer
Now, we can plug in the values and calculate the total number of days required to open the safe.

1. Calculate total combinations:
\[ 10^5 = 100000 \]

2. Calculate total time in seconds:
\[ 100000 \times 5 = 500000 \text{ seconds} \]

3. Convert total time to hours:
\[ \frac{500000}{3600} \approx 138.89 \text{ hours} \]

4. Calculate total days:
\[ \frac{138.89}{13} \approx 10.69 \text{ days} \]

Since we cannot have a fraction of a day in practical terms, we round up to the nearest whole number:

\[ \text{Total days} \approx 11 \]

Thus, the number of days sufficient enough to open the safe is **11 days**.