GreenEdge Mower Choice
A student runs a lawn-mowing business called GreenEdge. She has 160 lawns contracted for the season and charges $25 per lawn. A petrol mower costs $480 and uses fuel and maintenance worth $4.20 per lawn. A battery mower costs $1,150 and uses electricity worth $1.10 per lawn, but after 90 lawns it needs a compulsory $180 battery service before it can be used again. If she mows 90 lawns or fewer, no battery service is needed. Both mowers have no resale value at the end of the season. GreenEdge wants a season profit of at least $2,000 after paying for the mower and its running costs. She can afford either mower's purchase price, but she wants the option that gives the higher profit at 160 lawns and the lower risk if the contracted lawns fall to 120. She also wants to know how many lawns would make the battery mower cheaper overall.
1. Write total cost expressions for the petrol mower and the battery mower for n lawns, including the battery service when n > 90. Calculate the total cost and profit for each at 160 lawns.
Read the worked answer
For $n$ lawns:
Petrol: $C_p = 480 + 4.20n$.
Battery: $C_b = 1150 + 1.10n$ when $n \le 90$, and $C_b = 1150 + 1.10n + 180$ when $n > 90$.
At $n=160$, the service applies.
$C_p = 480 + 4.20(160) = 480 + 672 = 1152$.
$C_b = 1150 + 1.10(160) + 180 = 1150 + 176 + 180 = 1506$.
Revenue is $25 \times 160 = 4000$.
Petrol profit: $4000 - 1152 = 2848$.
Battery profit: $4000 - 1506 = 2494$.
Both exceed $2,000, but the petrol mower gives $2848 - $2494 = $354 more profit at 160 lawns.
2. At 120 lawns, recalculate and decide which mower gives the lower risk. Does the answer change from 160 lawns?
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At $n=120$, the battery service still applies because $120 > 90$.
$C_p = 480 + 4.20(120) = 480 + 504 = 984$.
$C_b = 1150 + 1.10(120) + 180 = 1150 + 132 + 180 = 1462$.
Revenue is $25 \times 120 = 3000$.
Petrol profit: $3000 - 984 = 2016$.
Battery profit: $3000 - 1462 = 1538$.
The petrol mower still gives the higher profit, by $2016 - 1538 = $478. It also meets the $2,000 target, while the battery mower does not. The answer does not change; the petrol mower is the lower-risk choice at both 160 and 120 lawns.
3. Solve for the number of lawns at which the battery mower becomes cheaper overall, including the service. Interpret the result for GreenEdge's 160-lawn contract.
Read the worked answer
For more than 90 lawns, set the costs equal:
$480 + 4.20n = 1150 + 1.10n + 180$.
The right side is $1330 + 1.10n$, so
$480 + 4.20n = 1330 + 1.10n$
$3.10n = 850$
$n = 850/3.10 = 274.19\ldots$
Since the number of lawns must be whole, the battery mower becomes cheaper at 275 lawns. For 90 lawns or fewer the service does not apply, and solving $480 + 4.20n = 1150 + 1.10n$ gives $n = 216.1$, which is outside that range, so there is no cheaper switch below 90 lawns.
At GreenEdge's 160 lawns, the battery mower is not cheaper; the petrol mower saves $354. The battery mower would only be justified if the contract grew to at least 275 lawns.