In her Algebra II Class , Mrs. Pemdas writes the following statement on the board: " a varies inversely as the product of `b^2` and c , and directly as `d^3` . " She then asks her students to translate the statement into an equation . What of the following equations , with k as the constant of proportionality , is a correct translation of Mrs. Pemdas's statement ?

Options

A. `a=(kd^3)/(b^2c)`
B. `a=(kb^2c)/d^3`
C. `a=(b^2cd^3)/k`
D. `a=(b^2c)/(kd^3)`

Answer

To translate Mrs. Pemdas's statement into an equation, we need to break down the statement into its components and understand the relationships described.

### Step-by-step Solution:

1. **Understanding the Relationships**:
- The statement "a varies inversely as the product of \( b^2 \) and \( c \)" means that as the product \( b^2 \cdot c \) increases, \( a \) decreases. This can be expressed mathematically as:
\[ a \propto \frac{1}{b^2 \cdot c} \]

2. **Direct Variation**:
- The statement "and directly as \( d^3 \)" means that as \( d^3 \) increases, \( a \) also increases. This can be expressed as:
\[ a \propto d^3 \]

3. **Combining the Relationships**:
- Since \( a \) varies inversely with \( b^2 \cdot c \) and directly with \( d^3 \), we can combine these two relationships:
\[ a \propto \frac{d^3}{b^2 \cdot c} \]

4. **Introducing the Constant of Proportionality**:
- To convert the proportionality into an equation, we introduce a constant \( k \):
\[ a = k \cdot \frac{d^3}{b^2 \cdot c} \]

5. **Final Equation**:
- Rearranging gives us the final equation:
\[ a = \frac{k \cdot d^3}{b^2 \cdot c} \]

### Conclusion:
Thus, the correct translation of Mrs. Pemdas's statement into an equation is:
\[ a = \frac{k \cdot d^3}{b^2 \cdot c} \]
This corresponds to **Option 1**.