To determine where the decimal point should be placed in the product of \( 6 \times 3.01 \), we first calculate the exact product:
\[ 6 \times 3.01 = 18.06 \]
Now, let's consider the placement of the decimal point in this product:
- Dylan's claim: He suggests placing the decimal between eight and zero, which would give us \( 180.6 \).
- Lucas's claim: He suggests placing the decimal between one and eight, which would give us \( 18.06 \).
Next, we can use estimation to confirm our understanding of this multiplication:
- Estimate \( 3.01 \) as approximately \( 3 \).
- Then multiply:
\[ 6 \times 3 = 18 \]
This estimation suggests that we should expect a product close to \( 18 \).
Now let's analyze the place value:
- The product \( 18.06 \) shows that the decimal point moves two places to the left in comparison to \( 1806 \) (the product of \( 6 \times 301\) if we were to multiply the whole numbers). This indicates that with decimal arithmetic, we have to be mindful of where the decimal point goes based on the number of decimal places in the factors.
Since \( 3.01 \) has two decimal places (the tenths and hundredths), when we multiply it by \( 6 \), we also take into account the shift of the decimal point by two places in the final answer.
Conclusion:
- Dylan is incorrect: \( 180.6 \) would imply that no decimal point shift occurred.
- Lucas is correct: \( 18.06 \) properly reflects the decimal placing derived from the factors \( 6 \) and \( 3.01 \).
Thus, the correct placement of the decimal gives us \( \textbf{18.06} \).