Every 8 days a mass of a certain radioactive substance decreases to exactly one-half of its value at the beginning of the 8day period. If the initial amount of the radioactivce substance is 75 grams, which equation gives the number of grams in the mass, M,that remains after d days?

Options

A. `M=75((d)/(16))`
B. `M=75((8)/(d))^(2)`
C. `M=75((1)/(2))^(8d)`
D. `M=75((1)/(2))^((d)/(8))`

Answer

To solve the problem, we need to find an equation that represents the mass \( M \) of a radioactive substance after \( d \) days, given that it halves every 8 days and starts with an initial mass of 75 grams.

### Step-by-Step Solution:

1. **Identify the Initial Mass**:
The initial mass of the radioactive substance is given as 75 grams.
\[ M_0 = 75 \text{ grams} \]

2. **Determine the Halving Period**:
The problem states that the mass decreases to half every 8 days. This means that after 8 days, the mass will be:
\[ M_1 = \frac{M_0}{2} = \frac{75}{2} \text{ grams} \]

3. **Generalize the Halving Process**:
After every 8 days, the mass halves. Therefore, after \( n \) periods of 8 days, the mass can be expressed as:
\[ M_n = M_0 \left( \frac{1}{2} \right)^n \]
where \( n \) is the number of 8-day periods that have passed.

4. **Relate \( n \) to \( d \)**:
The number of 8-day periods that fit into \( d \) days can be calculated as:
\[ n = \frac{d}{8} \]

5. **Substitute \( n \) into the Mass Equation**:
Now we can substitute \( n \) into the mass equation:
\[ M = 75 \left( \frac{1}{2} \right)^{\frac{d}{8}} \]

6. **Final Equation**:
Therefore, the equation that gives the number of grams \( M \) that remains after \( d \) days is:
\[ M = 75 \left( \frac{1}{2} \right)^{\frac{d}{8}} \]

### Summary:
The equation that represents the remaining mass of the radioactive substance after \( d \) days is:
\[ M = 75 \left( \frac{1}{2} \right)^{\frac{d}{8}} \]