Question

The function f(x) = 15(2)x represents the growth of a frog population every year in a remote swamp. Elizabeth wants to manipulate the formula to an equivalent form that calculates every half-year, not every year. Which function is correct for Elizabeth's purposes?

f(x) = 15(2 to the one half power)2x

f(x) = 15(2^2)^(x/2)

f(x) = fifteen halves(2)x

f(x) = 30(2)^x

Answer 1

The given function can also be written in the form:

f (x) = 15 (1 + 1)^x

Where this equation is similar to the general growth rate equation:

f (x) = k (1 + i)^n

Where,

i = percent rate of growth

n = number of years

Since we want to set that n = x/2, therefore we need to find the new i. We can do this by equating the two:

(1 + i)^x/2 = (2)^x

Multiplying the exponents of both sides by 1/x:

(1 + i)^1/2 = 2

1 + i = 4

i = 3

Therefore the new equation is:

f (x) = 15 (4)^(x/2)

or

f (x) = 15 (2^2)^(x/2) (ANSWER)