Question

Lionfish are an invasive species, with an annual growth rate of 69%. A scientist guesses there are 9,000 lionfish in a body of water after the first year.

Part A: Write the explicit equation for f(n) that represents the number of lionfish in the bay after n years. SHOW ALL WORK.

Part B: How many lionfish will be in the bay after 6 years? SHOW ALL WORK AND ROUND TO NEAREST WHOLE NUMBER.

Part C: If scientists remove 1,400 fish per year from the bay after the first year, what is the recursive equation for f(n)? SHOW ALL WORK.

Answer 1

Answer & Step-by-step explanation:

Part A: The explicit equation for f(n) that represents the number of lionfish in the bay after n years can be written as:

f(n) = 9000 * (1 + 0.69)^n

where 9000 is the initial number of lionfish, 0.69 is the growth rate, and n is the number of years.

Part B: To find the number of lionfish in the bay after 6 years, we substitute n = 6 into the explicit formula from Part A and simplify:

f(6) = 9000 * (1 + 0.69)^6 = 9000 * (1.69)^6 = 9000 * 11.34 = 102,060.5

Rounding to the nearest whole number, there will be approximately 102,061 lionfish in the bay after 6 years.

Part C: The recursive equation for f(n) can be found by subtracting 1,400 from the previous year's population and then applying the annual growth rate. Thus, we have:

f(1) = 9000 f(n) = f(n-1) - 1400 + 0.69f(n-1) = 0.69f(n-1) - 1400

where f(1) is the initial number of lionfish and n represents the number of years after the first year.