Q:

(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.

Accepted Solution

A:
Answer:a=55b=5T=17Step-by-step explanation:The general form of the equation is:[tex]N(t)=ab^{\frac{t}{T}}[/tex]For t = 0:[tex]N(0)=ab^{\frac{0}{T}}\\N(0) = a = 55\\a=55[/tex]Since there has been a fivefold increase after 17 years, at t = 17, N(17) = 55*5[tex]N(17)=55b^{\frac{17}{T}}\\55*5 = 55b^{\frac{17}{T}}\\b^{\frac{17}{T}} = 5[/tex]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:[tex]b^{\frac{t}{T}}= 5^{\frac{17n}{17}}[/tex]Therefore, the function that represents N(t) is:[tex]N(t)=55*5^{\frac{t}{17}}[/tex]