Answered

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 the total DIY investment cost: \(\$250 + \$135 + \$69 + \$25 = \$479\).
  • Define the professional service cost over \(m\) months: \(P(m) = 75 + 45(m-1)\).
  • Set up the inequality \(75 + 45(m-1) \ge 479\) and solve for \(m\).
  • Determine the minimum integer \(m = 10\).
  • The final answer is \(\boxed{10}\).

Explanation

  1. Calculating the DIY Investment Cost

To determine when the cost of the professional lawn care service exceeds the initial cost of purchasing the equipment for DIY yard maintenance, we must first calculate the total investment required for the DIY option. The Hendersons' initial equipment expenses are the sum of the costs for the lawnmower, edger, hedge trimmer, and rake:
\[\$250 + \$135 + \$69 + \$25 = \$479.\]
This represents the total fixed cost the Hendersons would incur immediately if they choose to do the lawn work themselves.

  1. Formulating the Professional Service Cost Function

Next, we define the cost of the professional service as a function of the number of months, \(m\). The first month costs \(\$75\), and every subsequent month (i.e., for \(m-1\) months) costs \(\$45\). Therefore, the total cost \(P(m)\) after \(m\) months is:
\[P(m) = 75 + 45(m-1)\]
for any integer \(m \ge 1\). We want to find the smallest number of months \(m\) for which the professional service cost meets or exceeds the DIY equipment cost of \(\$479\), which is when the Hendersons would effectively "recover" their initial DIY investment through professional savings:
\[75 + 45(m-1) \ge 479.\]

  1. Solving for the Break-Even Point

Now, we solve for \(m\):
\[45(m-1) \ge 479 - 75\]
\[45(m-1) \ge 404\]
Dividing both sides by 45:
\[m - 1 \ge \frac{404}{45} \approx 8.977...\]
Adding 1 to both sides:
\[m \ge 9.977...\]
Since \(m\) must be an integer number of months, we round up to the nearest whole number. Thus, \(m = 10\). After 10 months, the total cost of professional services will be \(75 + 45(10-1) = 75 + 405 = \$480\), which is greater than the \(\$479\) spent on the equipment.

  1. Conclusion and Final Answer

We have determined that it takes 10 months for the cumulative cost of the professional service to exceed the total initial investment of the DIY equipment. Comparing this result with the provided options (5, 9, 10, 6 months), we find that 10 months is the correct choice, which is option (c). Thus, the Hendersons recover their DIY equipment costs relative to the professional service cost after 10 months. The final answer is \(\boxed{10}\).

Examples

Deciding between a professional service and doing a task yourself is a common financial dilemma. Whether it is lawn care, car maintenance, or home repairs, calculating the "break-even point"—the time it takes for the savings from DIY work to surpass the initial cost of tools—helps you make informed decisions. By using simple algebraic expressions like \(P(m) = C_{initial} + C_{recurring}(m-1)\), you can quickly evaluate long-term costs and determine if investing in equipment is a wise financial choice for your household budget.

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