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Nicholas sent a chain letter to his friends, asking them to forward the letter to more friends. Every 12 weeks, the number of people who receive the email increases by an additional 99%, and can be modeled by a function, P, which depends on the amount of time, t (in weeks). Nicholas initially sent the chain letter to 50 friends. Write a function that models the number of people who receive the email t weeks since Nicholas initially sent the chain letter. P(t)=◻

Nicholas sent a chain letter to his friends, asking them to forward the letter to more friends. Every 1212 weeks, the number of people who receive the email increases by an additional 99% 99 \% , and can be modeled by a function, P P , which depends on the amount of time, t t (in weeks).\newlineNicholas initially sent the chain letter to 5050 friends.\newlineWrite a function that models the number of people who receive the email t t weeks since Nicholas initially sent the chain letter.\newlineP(t)=□ P(t)=\square

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Q. Nicholas sent a chain letter to his friends, asking them to forward the letter to more friends. Every 1212 weeks, the number of people who receive the email increases by an additional 99% 99 \% , and can be modeled by a function, P P , which depends on the amount of time, t t (in weeks).\newlineNicholas initially sent the chain letter to 5050 friends.\newlineWrite a function that models the number of people who receive the email t t weeks since Nicholas initially sent the chain letter.\newlineP(t)=□ P(t)=\square
  1. Identify initial value and growth rate: Identify the initial value aa and the growth rate rr.\newlineThe initial number of friends Nicholas sent the chain letter to is 5050, so a=50a = 50.\newlineThe growth rate every 1212 weeks is an additional 99%99\%, which means the number of people increases to 199%199\% of the previous amount every 1212 weeks. To express this as a growth factor for the function, we convert the percentage to a decimal. So, r=99%=0.99r = 99\% = 0.99.
  2. Calculate growth factor: Calculate the growth factor (bb).\newlineThe growth factor is 11 plus the growth rate. Since the growth rate is 0.990.99, we add this to 11 to get the growth factor.\newlineb=1+rb = 1 + r\newlineb=1+0.99b = 1 + 0.99\newlineb=1.99b = 1.99
  3. Write the function: Write the function using the initial value and the growth factor.\newlineThe function that models the number of people who receive the email t t weeks since Nicholas initially sent the chain letter is in the form P(t)=a(b)(t/k) P(t) = a(b)^{(t/k)} , where k k is the period of the growth cycle, which is 12 12 weeks in this case.\newlineSubstitute 50 50 for 'a a ', 1.99 1.99 for 'b b ', and 12 12 for 'k k ' into the function.\newlineP(t)=a(b)(t/k) P(t) = a(b)^{(t/k)} 00

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