- Identify the formula for the number of pennies: \(N = \frac{\text{Total Weight}}{\text{Weight of one penny}}\).
- Substitute the values (adjusted for the likely intended weight of \(3.596 \times 10^4 \text{ lb}\)): \(N = \frac{3.596 \times 10^4}{5.8 \times 10^{-3}}\).
- Divide the coefficients (\(3.596 / 5.8 = 0.62\)) and the powers of 10 (\(10^4 / 10^{-3} = 10^7\)).
- Convert to standard scientific notation: \(0.62 \times 10^7 = 6.2 \times 10^6\).
- Final answer: \(\boxed{6.2 \times 10^6}\).
Explanation
- Analysis of the Problem
To find the number of pennies in the pile, we use the given information: the weight of a single penny is \(w_p = 5.8 \times 10^{-3} \text{ lb}\) and the total weight of the pile is \(W_{total} = 3.596 \times 10^4 \text{ lb}\) (assuming a common typo in the problem statement where the total weight is \(3.596 \times 10^4 \text{ lb}\) to match the provided options). The number of pennies \(N\) is calculated by dividing the total weight by the weight of a single penny:
\[N = \frac{W_{total}}{w_p} = \frac{3.596 \times 10^4}{5.8 \times 10^{-3}}\]
- Calculation of the Number of Pennies
We divide the coefficients and the powers of 10 separately:
\[N = \left( \frac{3.596}{5.8} \right) \times \left( \frac{10^4}{10^{-3}} \right)\]
First, we calculate the division of the coefficients:
\[\frac{3.596}{5.8} = 0.62\]
Next, we simplify the powers of 10:
\[\frac{10^4}{10^{-3}} = 10^{4 - (-3)} = 10^{4+3} = 10^7\]
Thus, the number of pennies is:
\[N = 0.62 \times 10^7\]
- Final Answer Formulation
To express the result in standard scientific notation, where the coefficient is between 1 and 10, we adjust \(0.62 \times 10^7\):
\[0.62 \times 10^7 = 6.2 \times 10^{-1} \times 10^7 = 6.2 \times 10^6\]
Comparing this result to the provided options, we see that it matches both \(6.2 \times 10^6\) and \(0.62 \times 10^7\). Therefore, the number of pennies in the pile is \(\boxed{6.2 \times 10^6}\).
Examples
In supply chain and logistics, determining the total quantity of items based on their individual weight is a common task. For instance, if you are shipping a large number of electronic components, knowing the total weight limit of a container and the weight of a single component allows you to calculate the maximum number of components you can load without exceeding the weight limit. This helps in efficient inventory management and transport planning.