The table shows the population of Arizona Bark Scorpions and Emperor Scorpions over a number of months. One species grows linearly, and the other grows exponentially. We need to determine which species grows exponentially, complete the table for months 3 and 4, and give the equation for the population growth of the Emperor Scorpion.

1. Problem Description

The table shows the population of Arizona Bark Scorpions and Emperor Scorpions over a number of months. One species grows linearly, and the other grows exponentially. We need to determine which species grows exponentially, complete the table for months 3 and 4, and give the equation for the population growth of the Emperor Scorpion.

2. Solution Steps

a) Determine which species grows exponentially.

The Arizona Bark Scorpion population is 500 at month 0, 750 at month 1, and 1000 at month

2. The population increases by 250 each month. This suggests a linear growth pattern.

The Emperor Scorpion population is 320 at month 0, 480 at month 1, and 720 at month

2. Let's examine the ratios:

$480/320 = 1.5$

$720/480 = 1.5$

Since the ratio between consecutive terms is constant, the Emperor Scorpion population grows exponentially.

b) Complete the table for months 3 and

4.

Since the Arizona Bark Scorpion grows linearly with an increase of 250 per month, we can calculate the population for months 3 and 4:

Month 3: $1000 + 250 = 1250$

Month 4: $1250 + 250 = 1500$

Since the Emperor Scorpion grows exponentially with a growth factor of 1.5, we can calculate the population for months 3 and 4:

Month 3: $720 * 1.5 = 1080$

Month 4: $1080 * 1.5 = 1620$

c) Give the equation for the population growth of the Emperor Scorpion.

The general equation for exponential growth is:

$P(t) = P_0 * r^t$

Where:

$P(t)$ is the population at time $t$

$P_0$ is the initial population

$r$ is the growth rate

$t$ is the time (in months)

In this case, the initial population $P_0 = 320$ and the growth rate $r = 1.5$.

Therefore, the equation for the population growth of the Emperor Scorpion is:

$P(t) = 320 * (1.5)^t$

3. Final Answer

a) Emperor Scorpion

b)

Month 3: Arizona Bark = 1250, Emperor = 1080

Month 4: Arizona Bark = 1500, Emperor = 1620

c) $P(t) = 320 * (1.5)^t$