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A shoe factory produces $1.81 \times 10^4$ pairs of shoes each month. Estimate how many pairs of shoes the factory will produce if it maintains that rate for 10 years. (There are $1.2 \times 10^2$ months in 10 years.)


  1. Estimate each coefficient:

$\begin{array}{l}
1.81 \approx 2 \\
1.2 \approx 1
\end{array}$


  1. Write the product:

$\left(2 \times 10^4\right)\left(1 \times 10^2\right)$



The shoe factory would make approximately $\square \times 10^6$ pairs of shoes in 10 years.

Verified Answer :

  • Estimate the monthly production rate as \(2 \times 10^4\) and the duration as \(1 \times 10^2\) months.
  • Multiply these values to find the total: \((2 \times 10^4) \times (1 \times 10^2)\).
  • Use the rule \((a \times 10^m) \times (b \times 10^n) = (ab) \times 10^{m+n}\) to compute the product.
  • The total estimated production is \(\boxed{2} \times 10^6\) pairs of shoes.

Explanation

  1. Analysis of the Production Estimation Problem

The shoe factory has a monthly production rate of \(1.81 \times 10^4\) pairs of shoes. We are asked to estimate the total number of shoes produced over 10 years. We are given that 10 years is equivalent to \(1.2 \times 10^2\) months. To estimate the total, we use the provided approximations: the monthly production rate \(1.81 \times 10^4\) is approximated as \(2 \times 10^4\), and the duration \(1.2 \times 10^2\) months is approximated as \(1 \times 10^2\).

  1. Calculating the Estimated Total Production

We now perform the estimation by multiplying the approximated monthly rate by the approximated number of months:
\[(2 \times 10^4) \times (1 \times 10^2)\]
According to the properties of exponents, we multiply the coefficients and add the exponents of the base 10:
\[(2 \times 1) \times 10^{4+2} = 2 \times 10^6\]
Thus, the factory is estimated to produce \(2 \times 10^6\) pairs of shoes over the 10-year period.

Examples

Estimating production is a common task in business planning and supply chain management. For instance, if a company wants to project its annual revenue or storage needs for the next few years, it will use estimates of its current production or sales rates combined with time frames. This allows managers to prepare logistics, secure raw materials, and plan for warehouse space efficiently without needing to perform exact calculations for every single unit produced. Simple power-of-ten estimations provide quick, reliable benchmarks for long-term strategic decision-making.

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