A declining income against rising costs
A fisherman purchases a boat for 1000 dollars at the beginning of 2011. Annual operating cost is 120 dollars in 2011 and increases by 40 dollars every year.
(i) Show total cost including purchase and operations through year is . mark
Annual fishing income is 500 dollars in 2011 and decreases by 10% yearly due to overfishing.
(ii) Show cumulative fishing income never exceeds 5000 dollars, however many years pass. marks
The boat can be sold for 600 dollars in 2011; sale price drops by 10% yearly. It is sold at the end of year , after fishing income is collected.
(iii) Find total income from fishing and boat sale in terms of . marks
The fisherman sells in the final year when this total income exceeds total cost.
(iv) Find that year. marks
[2014/CJC/JC1/mid-year/7]
Solution
Total cost through year n
Find the total cost through year , including the boat purchase.
Hint
Do not include the 1000 purchase in every annual AP term.
Answer · (i)
dollars.
Identify the AP
Year 1 is 2011 and costs 120 to operate.
Simplify
Separate the fixed cost from the quadratic running cost.
Upper bound on fishing income
Show that fishing income never exceeds 5000 dollars.
Hint
The income approaches 5000 but does not attain it in a finite year.
Answer · (ii)
dollars.
Sum n annual incomes
The first 500-dollar income has exponent 0.
Prove the strict ceiling
For every finite positive n,0.9^n is positive.
Fishing plus boat sale
Find total income when the boat is sold at the end of year .
Hint
The sale-price exponent is n−1, whereas the partial-sum formula contains 0.9^n.
Answer · (iii)
dollars.
Index the boat value
In 2011 its value is 600, so year n has n−1 decreases.
Add the two income sources
Do not sum the boat sale prices as though a boat were sold every year.
Last profitable sale year
Find the last profitable sale year.
Hint
An adjacent sign change needs a reason that later profit cannot return.
Answer · (iv)
.
Form total profit
The condition is strict: equality would not qualify.
Check the last sign change
Use full precision for the comparisons.
Exclude any later return to profit
For n≥8, the next income increment is at most 390(0.9)^7<440, while the next cost increment is 40n+120≥440. Thus profit keeps decreasing.
Map the index to the year
2011 is year 1, so add n−1.
Explore the idea
Separate a recurring income from a one-off sale
The question asks for the final profitable year, not the first one.