The size N , in thousands, of a certain animal population after years is given in the table below. By what percentage can the population be expected to grow over any 6-year period?

A) $48.67 \%$
B) $48 \%$
C) $58.69 \%$
D) $60.59 \%$

Question

The size N , in thousands, of a certain animal population after  years is given in the table below. By what percentage can the population be expected to grow over any 6-year period?

A) $48.67 \%$ 
B) $48 \%$ 
C) $58.69 \%$ 
D) $60.59 \%$

Quizplus · Accepted Answer

First, we can find the population after 6 years by looking at the table. The population in the 11th year is 49, so the population in the 6th year would be approximately halfway between the populations in the 5th and 6th years, which are 37 and 62 respectively. Thus, we can estimate the population after 6 years as 49 + (62-37)/2 = 49 + 12.5 = 61.5 (in thousands).

Next, we can use the formula for percentage growth:

percentage growth = ((new population - old population) / old population) x 100%

For a 6-year period, the old population is 49 and the new population is 61.5, so:

percentage growth = ((61.5 - 49) / 49) x 100% ≈ 25.51%

However, we need to be careful since this is only an estimate based on the midpoint approximation. To get a more accurate answer, we can use the formula for exponential growth:

N = N0 x e^(rt)

where N0 is the initial population, r is the growth rate (in decimal form), and t is the time elapsed (in years). We can use this formula to find the growth rate over a 6-year period by solving for r with N0 = 49, t = 6, and N = 61.5:

61.5 = 49 x e^(6r)
1.255102 = e^(6r)
ln(1.255102) = 6r
r ≈ 0.058939

This means that the population is growing at a rate of 5.8939% per year. Over a 6-year period, the overall percentage growth is:

percentage growth = ((1 + r)^6 - 1) x 100%
= ((1 + 0.058939)^6 - 1) x 100%
≈ 58.69%

Therefore, the answer is C, 58.69%.

The Size N , in Thousands, of a Certain Animal 48.67%48.67 \%48.67%

The Size N , in Thousands, of a Certain Animal $48.67 \%$