In January 2010 the price of tea in Flotinnia was 8180 per p

In January 2010, the price of tea in Flotinnia was $81.80 per pound.^1 Each year, the price of tea in Flotinnia increased by 3.4% of its value the previous year. Write an expression for the price of tea per pound in Flotinnia in January 2011. Is it possible to write this expression so that it has only one term (that is, so that the expression is not a sum of multiple terms)? Write expressions for the price of tea per pound in Flotinnia in January 2012 and January 2013. Let p(t) represent the price in dollars of a pound of tea in Flotinnia t years after January 2010. Write an equation that shows how p(t) is related to p(t - 1) for a given value of t. (Such an equation is called a recursive formula for p(t), since it defines p(t) in terms of other function values.) Now, write a formula for p(t) in terms of t only; that is, without referring to p(t - 1) or any other value of the function p. (Such a formula is called an explicit formula for p(t), since it defines p(t) directly in terms of the input variable t.) Explain how a student unfamiliar with exponential functions could come up with the formula you wrote in part (d) as a natural generalization of the calculations performed in parts (a) and (b).

Solution

If the current price of tea in Flotinnia is $ x per lb, then next year, the increase in its price is 3.4 % x = 0.034x so that the price of tea in the next year will be x + 0.034x = $ (1.034)x per lb.

(a).In January, 2010, the price of tea in Flotinnia is $ 81.80 per lb. Hence, in January,2011, the price of tea in Flotinnia will be $ 81.80*1.034 (= $ 84.58 per lb, on rounding off to the nearest cent). It is possible to write the expression as only one term which will be 1.034 times the previous year’s price.

(b). In January,2012, the price of tea in Flotinnia will be $ 81.80*1.034*1.034 = $ 87.46 per lb ( on rounding off to the nearest cent and in January,2013, the price of tea in Flotinnia will be $ 81.80*1.034*1.034*1.034 = $ 90.43 per lb ( on rounding off to the nearest cent).

(c) If p(t) is the price of tea in Flotinnia, t years after January, 2010, then p(t) = p(t-1)*1.034.

(d) If p(t) is the price of tea in Flotinnia, t years after January, 2010, then p(t)=$ 81.80(1.034)^t per lb.

(e). If the current price of tea in Flotinnia is $ x per lb, then next year, the increase in its price is 3.4 % x = 0.034x so that the price of tea in the next year will be x + 0.034x = $ (1.034)x per lb. This means that the previous year’s price is multiplied by 1.034. For the price of tea, 2 years from the current year, the current price gets multiplied by 1.034*1.034 = (1.034)². Thus, for the price of tea, t years from now, the current price gets multiplied by 1.034*1.034*1.034…t times = (1.034)^t