Final answer:
The function N(t) = 42 (1.75)^t indicates a 75% yearly growth rate in the number of branches, with the branches increasing by that percentage each year. The percentage is found by calculating (1.75 - 1) * 100%. If desired, the time it takes for a certain multiplication of the branches can be calculated using logarithmic equations.
Step-by-step explanation:
The question is about calculating the yearly percent change in the number of branches represented by the function N(t) = 42 (1.75)t, where t is time in years. To determine the percentage growth rate, we recall that percentage growth is represented by the formula (new amount - original amount) / original amount * 100%. In this case, assuming t = 1 to find the annual growth, we subtract the base value (1) from 1.75 to get 0.75. To convert this to a percentage, we multiply by 100 to get a 75% yearly growth rate. This means that the number of branches increases by 75% each year.
If we want to reverse-engineer and determine the time it takes for the branches to triple (M = 3), we can use a logarithmic equation derived from the basic exponential growth formula, which reveals the number of periods required for growth at a given rate. For example, using Eq. 1.5, if the base (b) is 1.05, then Multiplying the growth rate by 100 gives us the percentage rate (5%). Following the example where M = 3, we would solve for n (the number of steps or years in this case), and then multiply the outcome by the annual growth rate to find the number of years needed for the branches to triple.