Final answer:
To find when the population doubles from a given number, use the exponential growth formula and set N(t) to 2N0. With a doubling time of 10 years, t equals the doubling time T. To reach 7 billion from 100 people, solve 100 * (2^n) = 7,000,000,000.
Step-by-step explanation:
To solve the initial value problem and determine when the population doubles, we look at the example town starting with 100 people in 1900 and doubling every 10 years. To find when this population reaches 7 billion, we note it took about 260 years to do so, according to the text. However, to calculate the exact number of years for the population to double from any given number, we can use the formula for exponential growth, where N(t) = N0 * (2^(t/T)), with N(t) being the population at time t, N0 the initial population, and T the doubling time.
To find when the population has doubled, we set N(t) = 2N0 and solve for t. In this case, with a doubling time of 10 years, we simply have t = T, so the population doubles every 10 years.
If we want to know after how many years the population would become 7 billion starting from 100, we need to find n where 100 * (2^n) = 7,000,000,000. To solve for n, we can take the logarithm of both sides which gives us n = log2(7,000,000,000/100). Once we have n, we can multiply by the doubling time (10 years) to find the total number of years required.