92 MATHEMATICAL THEORY OF INVESTMENT 5. On a certain railroad there is a grade which necessitates the use of a " helper " engine and two crews. If the cost of wages, fuel, oil, and repairs is $10,000 per annum, and a new engine costing $12,000 must be purchased every 20 years, how much could the company afford to expend in reducing the grade to the point where the helper could be dispensed with, money worth 4% ? 6. A building costs $100,000 to erect. Annual repairs cost $500. Every 10 years it* must be thoroughly overhauled at a cost of $5000, and every 75 years it must be rebuilt at the original cost. What amount of endow- ment would be necessary in order to build and maintain the structure indefinitely, if money is worth 4% ? 7. What amount can be expended in treating a telegraph pole costing $3, to extend its life from 8 to 15 years, if the cost of setting the pole is $5, on .the supposition that money is worth 5% ? 8. Prove that the amount to be expended in doubling the life of an article is ^ ^ ^ where k years is the life of the article without the added expenditure. (Obtain proof by means of formula (8).) 9. If an article of cost C has at the end of its period of service a scrap value S, prove that the amount that may be expended in extending its life from k years to TC' years is given by the formula x=vk'-':s^l-(c-s) m ^ 10. A man buys a team of horses for $500 and a wagon and harness for $150. The team will have to be replaced in 10 years and the wagon and harness in 20 years. Feed and blacksmith bills cost $200 a year, taxes and insurance $20, and wages of the driver $600. How much must he make in a year in order to realize 8% on his investment, if the investment is capitalized at 5% ? 35. Continuous annuities. We may think of an annuity for which the total amount paid in a year is a fixed sum, but the payments are infinitesi- mally small and are made momently. Such a hypothetical annuity is called a continuous annuity. Such annuities do not exist in the actual business world, but they are approximated by the business of large concerns which are receiving many small sums every day. PROBLEM.To find the amount of a continuous annuity of 1 per annum for n years. We will denote the amount of a continuous annuity by s-yy and the present value by a,T We have T (P) 1. ('l+l')"——! .^ = lim .^p) = hm -^———L———. p= eo p ==co — j>[(i+.y-i]