Question

(1 point) The number of households with cable TV service in a certain community, N, begins at only 55 households and has seen a fivefold increase every 17 years. Give the constants a, b, and T so that N is represented by a function of the form N=abt/T, where t is the time in years since N was first measured.

Answer 1

a=55

b=5

T=17

Explanation:

The general form of the equation is:

`N(t)=ab^{(t)/(T)}`

For t = 0:

`N(0)=ab^{(0)/(T)}\\N(0) = a = 55\\a=55`

Since there has been a fivefold increase after 17 years, at t = 17, N(17) = 55*5

`N(17)=55b^{(17)/(T)}\\55*5 = 55b^{(17)/(T)}\\b^{(17)/(T)} = 5`

If at every 17*n years there in an increase of 5^n, one can deduct that the values for T and b are respectively 17 and 5:

`b^{(t)/(T)}= 5^{(17n)/(17)}`

Therefore, the function that represents N(t) is:

`N(t)=55*5^{(t)/(17)}`