The count rate from `100cm^(3)` of a radioactive liquid is `c`. Some of this liquid is now discarded. The count rate of the remaining liquid is found to be `c//10` after three half-lives. The volume of the remaining liquid, in `cm^(3)`, is

Options

A. 20
B. 40
C. 60
D. 80

Explanation

To solve the problem step by step, we can follow this approach:

### Step 1: Understand the Initial Conditions
We start with a volume of `100 cm³` of a radioactive liquid, which has an initial count rate of `C`.

**Hint:** Identify the initial conditions and the relationship between volume and count rate.

### Step 2: Determine the Count Rate per cm³
The count rate per cubic centimeter can be calculated as:
\[ \text{Count rate per cm³} = \frac{C}{100} \]

**Hint:** Calculate the count rate for a smaller volume to understand how the total count rate distributes.

### Step 3: Count Rate After Three Half-Lives
After three half-lives, the count rate of the remaining liquid is given to be `C/10`. The general formula for the count rate after `n` half-lives is:
\[ \text{Count rate after } n \text{ half-lives} = \frac{C}{2^n} \]
For three half-lives (`n = 3`):
\[ \text{Count rate after 3 half-lives} = \frac{C}{2^3} = \frac{C}{8} \]

**Hint:** Use the half-life formula to find the new count rate after a specified number of half-lives.

### Step 4: Relate the Count Rate of the Remaining Liquid
Let the volume of the remaining liquid be `V cm³`. The count rate of the remaining liquid can also be expressed as:
\[ \text{Count rate of remaining liquid} = V \times \frac{C}{100} \]
After three half-lives, we set this equal to the new count rate:
\[ V \times \frac{C}{100} = \frac{C}{10} \]

**Hint:** Set up an equation using the count rate of the remaining liquid and the new count rate after decay.

### Step 5: Solve for Volume `V`
Now, we can simplify the equation:
\[ V \times \frac{C}{100} = \frac{C}{10} \]
Dividing both sides by `C` (assuming `C` is not zero):
\[ V \times \frac{1}{100} = \frac{1}{10} \]
Multiplying both sides by `100` gives:
\[ V = 100 \times \frac{1}{10} = 10 \]

**Hint:** Isolate `V` to find the volume of the remaining liquid.

### Step 6: Correct the Calculation
Since we need to consider that the count rate after three half-lives should equal `C/10`, we need to equate:
\[ V \times \frac{C}{800} = \frac{C}{10} \]
Now, simplifying:
\[ V = \frac{C/10}{C/800} = \frac{800}{10} = 80 \]

### Final Answer
The volume of the remaining liquid is:
\[ \boxed{80 \, cm^3} \]