Farmer Buck tasked William, his financial consultant, with creating a mathematical model that explains the details of ranching.

Assume the following:

$1.$ Definitions: A heifer is an unbred cow. Herd size is adult pregnant female cows $+$ adult nonpregnant female cows $($does not include calves or bulls$)$ and in other words, means the minimum population of cows Farmer Buck needs to maintain for his given goals throughout the year $.$ A replacement cow is a one$-$year$-$old cow you receive when you trade your heifer to the auction house from a different ranch and a different father$-$bull $($there is no cash inflow or cash outflow when trading a heifer for a replacement cow$).$ A bull is a male cow. Gross income is total income$/$revenue before any expenses.

$2.$ Model$/$formula: Assumes Farmer Buck has been in the ranching business for years already and only includes regular operational income and costs $($i$.$e$.$ no start$-$up costs, no purchases of cows, etc.$).$ Every $25$th bull cow in the herd requires an additional bull.

$3.$ Expenses: In the middle of October of every year, all unvaccinated cattle over $7$ months old are vaccinated. Calves under $7$ months old are not vaccinated. If a cow is pregnant$/$lactating$,$ that cow is not vaccinated. Vaccination fees are $$50$ per head $($charged only by the cattle to be vaccinated$).$ Yearly property tax is $$4,000$ and yearly property insurance is $$2,000$ independent of herd size. Farmer Buck spends $$500/$week on cow treats, independent of herd size, and $$500/$month on electricity, independent of herd size $($for pumping water, exterior illumination, etc.$).$

$4.$ Pastures: There are two pastures: the breeding pasture and the nursing pasture. The breeding pasture is for pregnant cows or lactating mothers with calves under $9$ months. Once the calf is $9$ months old, the calf and its mother will be moved from the nursery pasture to the breeding pasture. At twelve months old, the calf is traded for a one$-$year$-$old replacement cow. However, a calf will not become fertile until it is at least $15$ months. Since replacement cows from the auction are assumed to be $12$ months old, there is a three$-$month delay until the cow is fertile for breeding. When a replacement cow is brought to the ranch, she will be placed in the breeding pasture. There is a one$-$month delay for Farmer Buck to switch the cows from the nursing pasture to the breeding pasture.

$5.$ Gestation: The gestation $($pregnancy$)$ period of a cow is $10$ months, and a cow gives birth to only one calf at a time $($no twins$).$ The cow is bred $($impregnated$)$ in the middle of the month and delivers $($gives birth$)$ in the middle of the month. When a cow gives birth to a calf, she will not be bred again for the $9$ months she is lactating $($until the calf is weaned in $9$ months$).$ There is another one month delay for a bull to breed the cow. There is a $50\%$ chance of a calf, when born, being a bull or a heifer. Only non$-$pregnant, non$-$lactating cows are bred $($impregnated$).$ Calves have a $20\%$ mortality rate $($poison$,$ illness, etc.$).$ A cow is fertile for the first eight years of its life $($once it is $15$ months old, of course$),$ after which fertility rate drops to $20\%$ for the next four years, and then becomes infertile. A bull can mate for the first ten years of its life $($once it is $15$ months old, of course$)$ after which fertility rate drops to $20\%$ for the next two years, and then becomes sterile.

$6.$ Income: The market price Farmer Buck sells bulls to the auction house at is $$2,000.$ This is also the price that he may buy a bull at$.$ Selling bulls is the only source of income Farmer Buck has identified. Replacement cows are worth $$1,500$ for both sale and purchase, but Farmer Buck pays no cash for them, only trades$-$in his one$-$year$-$old heifers at a ratio of $1$:$1$ for an equal amount of one$-$year$-$old replacement cows.

QUESTION $1$: If Farmer Buck has a herd size of $5$ this year $($middle of the year$)$ in how many years will he have a herd size of $56?$