Question

the population of deer and wild boars on an island in the pacific ocean were recorded over a period of five years. the results are shown in the table. which statement best describes the results?

time (in years)	deer	boars
1	2420	3850
2	2540	4235
3	2860	4659
4	3080	5124
5	3300	5637

populations of deer and boars

• the population of deer is growing exponentially
• both populations are growing at a constant rate.
• both populations are growing at an exponential rate.
• the population of boars is growing exponentially.

Explanation:

1. First, analyze the deer population:

• Calculate the differences between consecutive years:
• Year 0 to 1: \(2420 - 2200 = 220\)
• Year 1 to 2: \(2640 - 2420 = 220\)
• Year 2 to 3: \(2860 - 2640 = 220\)
• Year 3 to 4: \(3080 - 2860 = 220\)
• Year 4 to 5: \(3300 - 3080 = 220\)
• The deer population has a constant difference (220) each year, so it's growing linearly (constant rate of change).

1. Next, analyze the boar population:

• Calculate the ratios between consecutive years:
• Year 0 to 1: \(\frac{3850}{3500}=1.1\)
• Year 1 to 2: \(\frac{4235}{3850}=1.1\)
• Year 2 to 3: \(\frac{4659}{4235}=1.1\)
• Year 3 to 4: \(\frac{5124}{4659}=1.1\)
• Year 4 to 5: \(\frac{5637}{5124}=1.1\)
• The boar population has a constant ratio (1.1) each year, so it's growing exponentially (constant multiplicative rate of change).

1. Now, evaluate the options:

• "The population of deer is growing exponentially" is incorrect because deer grow linearly.
• "Both populations are growing at a constant rate" is incorrect because boars grow exponentially (constant ratio, not constant difference).
• "Both populations are growing at an exponential rate" is incorrect because deer grow linearly.
• "The population of boars is growing exponentially" is correct as shown by the constant ratio in their population growth.

Answer:

The population of boars is growing exponentially.