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The Hendersons have just bought a home that requires some monthly yard maintenance. They are trying to decide if they should hire a professional lawn care service to maintain the property or do it themselves. Below are the costs associated with both options:




Professional Service Option		Do-It-Yourself Option

\$75 for the first month		\$250 lawnmower

\$45 per month after initial visit		\$135 edger

Includes mowing and edging the lawn, trimming the bushes, and raking the leaves.		\$69 hedge trimmer

\$25 rake






How long would it take before the Hendersons would recover their costs if they chose to do it themselves over the professional service?


A5 months
B9 months
C10 months
D6 months

Verified Answer :

  • Calculate total DIY cost: \(\$250 + \$135 + \$69 + \$25 = \$479\).
  • Define professional service cost function: \(C_P(n) = 45n + 30\).
  • Set up inequality to find break-even point: \(45n + 30 \geq 479\).
  • Solve for \(n\): \(n \geq 9.98\), meaning the costs are recovered after \(\boxed{10}\) months.

Explanation

  1. Calculating Total DIY Investment Cost

To determine when the Hendersons will recover the costs of their DIY yard equipment, we first calculate the total initial investment required for the DIY option. Summing the costs of the equipment:
\[\text{Total DIY cost} = \$250 (\text{lawnmower}) + \$135 (\text{edger}) + \$69 (\text{hedge trimmer}) + \$25 (\text{rake}) = \$479.\]

  1. Defining the Professional Service Cost Function

Next, we define the cumulative cost of the professional service over \(n\) months. The cost is \(\$75\) for the first month and \(\$45\) for each subsequent month. Therefore, the total cost for \(n\) months, denoted as \(C_P(n)\), is given by the formula:
\[C_P(n) = 75 + 45(n - 1)\]
Expanding this expression, we get:
\[C_P(n) = 75 + 45n - 45 = 45n + 30.\]

  1. Determining the Break-Even Point

We need to find the number of months \(n\) for which the professional service cost equals or exceeds the initial DIY investment of \(\$479\). We set up the inequality:
\[45n + 30 \geq 479\]
Subtracting 30 from both sides:
\[45n \geq 449\]
Dividing by 45:
\[n \geq \frac{449}{45} \approx 9.98.\]
Since \(n\) must be a whole number of months, we round up to the nearest integer, which is \(10\). At \(n = 9\), the professional cost is \(45(9) + 30 = \$435\), which is less than the DIY investment. At \(n = 10\), the professional cost is \(45(10) + 30 = \$480\), which covers the initial cost of \(\$479\). Therefore, it takes \(10\) months to recover the costs.

Examples

This problem is a classic example of a break-even analysis, a fundamental concept in personal finance and business. By comparing the upfront investment of a DIY project against the recurring cost of a service, individuals can make informed financial decisions. For instance, if you were considering buying a coffee machine versus buying a coffee at a cafe daily, you could use the same approach to determine how many days it would take until the cost of the machine is paid off by the savings.

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