The function f (x) = 245 (4) ^x represents the growth of a fruit fly population every YEAR in the orange grove. Stacy wants to manipulate the formula to an equivalent form that calculates every MONTH, not every year. Which function is correct for Stacy's purposes?

a. f (x) = 45 (4) ^x

b. f (x) = 45 (4^12) x/12

c. f (x) = 2752 (4) ^x

d. f (x) = 245 (4^1/12) ^12x

Answer is f (x) = 245 (4^1/12) ^12x

Step-by-step explanation:

The general formula for the growth of population in years is:

f (x) = P0 (1+r) ^x

where 1+r is the rate of growth in this case 4 and x is time in years. To take it to months we do not change initial population, this doesn't vary by changing time scale (there are 45 fruit flies in time 0)

we change the time period x to 12 x because we want our formula to remain equivalent to the original one so if x = 1 year in months this would be equivalent to twelve months (12x = 12 with x = 1) taht is the same

However we need to reescale our growth rate to a value that calculates it by month growth. After 12 time periods of a month our grow rate should be equivalent to the growth rate of a year. So For that reason we elevate to a 1/12 because of exponent rules

(4^1/12) ^12x = (4^12x/12) because of exponent of an exponent is multiplying exponents

= (4) ^x we arrive to the same expression. So