Questions: What was the production cost in the year 2020? Write a formula for Q as a function of t, if (a) the initial amount is 23 and Q is increasing by f(Q)=23(1.12)^6 (b) the initial amount is 31 and Q is decreasing by Q=31(.88) t

Solution Steps

1. For part (a), we need to write a formula for \( Q \) as a function of \( t \) given the initial amount and the growth rate.
2. For part (b), we need to write a formula for \( Q \) as a function of \( t \) given the initial amount and the decay rate.

For the increasing function, we have the initial amount \( Q_0 = 23 \) and the growth rate \( r = 1.12 \). The formula for \( Q \) as a function of time \( t \) is given by:

\[ Q(t) = 23(1.12)^t \]

Substituting \( t = 6 \):

\[ Q(6) = 23(1.12)^6 \approx 45.398 \]

For the decreasing function, the initial amount is \( Q_0 = 31 \) and the decay rate is \( r = 0.88 \). The formula for \( Q \) as a function of time \( t \) is:

\[ Q(t) = 31(0.88)^t \]

Substituting \( t = 6 \):

\[ Q(6) = 31(0.88)^6 \approx 14.397 \]

Final Answer