Finding the Growth Rate Help

• The population of a country is growing exponentially. In the year 2000, it was 10 million and in 2005, it was 12 million. What is the growth rate? In the year t = 0 (2000), the population was 10 million, so n ₀ = 10. The growth formula becomes n ( t ) = 10 e ^rt . When t = 5 (the year 2005), the population is 12 million, so n ( t ) = 12. We will solve the equation 12= 10 e ^{5 r} for r .

The country’s population is growing at the rate of 3.6% per year.

• Suppose a bacteria culture contains 2500 bacteria at 1:00 and at 1:30 there are 6000. What is the hourly growth rate?

• Because we are asked to find the hourly growth rate, t must be measured in hours and not minutes. Initially, at t = 0, the population is 2500, so n ₀ = 2500. Half an hour later, the population is 6000, so t = 0.5 and n ( t ) = 6000. We will solve for r in the equation 6000 = 2500 e ^{0.5 r} .

The bacteria are increasing at the rate of 175% per hour.

• A certain species of fish is introduced in a large lake. Wildlife biologists expect the fish’s population to double every four months for the first few years. What is the annual growth rate?

• If n ₀ represents the fish’s population when first put in the lake, then it will double to 2 n ₀ after . The growth formula becomes . This equation has two unknowns, n ₀ and r , not one. But after we divide both sides of the equation by n ₀ , r becomes the only unknown.

The fish population is expected to grow at the rate of 208% per year.