A conservationist monitors the populations of two plant species, Species A and Species B, … — SAT

A conservationist monitors the populations of two plant species, Species A and Species B, in a forest reserve. The table below shows the population of each species, $P(t)$ and $Q(t)$ respectively, $t$ years after the monitoring began.

$t$ (years)	Species A population, $P(t)$	Species B population, $Q(t)$
$0$	$100$	$100$
$2$	$150$	$144$
$4$	$200$	$207.36$

The population of Species A can be modeled by a linear function, where $r_P$ represents the constant increase in the number of plants per year. The population of Species B can be modeled by an exponential function, where $r_Q\%$ represents the constant percent increase in the number of plants per year. What is the value of $r_P - r_Q$ ?

$5$ Answer
B
$6$
C
$19$
D
$30$

Answer

To find

r_P

, we calculate the constant amount by which the linear population

P(t)

increases each year. Over the 2-year interval from

t = 0

t = 2

, the population increases by

150 - 100 = 50

. Since the growth is linear, the annual increase is

r_P = \frac{50}{2} = 25

plants per year. To find

r_Q

, we write the exponential model in the form

Q(t) = 100(b)^t

, where

b

is the annual growth factor. Using the value at

t = 2

, we have

144 = 100(b)^2

, which simplifies to

b^2 = 1.44

. Taking the square root gives the annual growth factor

b = 1.2

. A growth factor of

1.2

corresponds to an annual growth rate of

20\%

, so

r_Q = 20

. The difference between the two values is

r_P - r_Q = 25 - 20 = 5

. This matches the option with value 5.

Step-by-Step Solution

Calculate the annual rate of change (

r_P

) for the linear population model

P(t)

r_P = 25

Since Species A's population grows linearly, its rate of change is constant. The population increases from 100 to 150 over 2 years, which is an increase of

150 - 100 = 50

plants. Dividing this by the 2-year interval gives an annual growth rate of

\frac{50}{2} = 25

plants per year.

Determine the annual growth factor and percentage growth rate (

r_Q

) for the exponential population model

Q(t)

r_Q = 20

Since Species B's population grows exponentially, its population after

t

years can be written in the form

Q(t) = a(b)^t

, where

a

is the initial population and

b

is the annual growth factor. Using the data in the table:

Q(0) = a = 100

and

Q(2) = 100(b)^2 = 144

. Solving for

b

gives

b^2 = 1.44

, which means

b = \sqrt{1.44} = 1.2

. An annual growth factor of

1.2

corresponds to a constant percent increase of

(1.2 - 1) \times 100\% = 20\%

per year. Thus,

r_Q = 20

Calculate the difference between the two rates (

r_P - r_Q

5

Subtracting

r_Q

from

r_P

yields

25 - 20 = 5

Key Concept

Identifying and interpreting linear and exponential growth models from data tables, and converting rate of change across different time intervals.

Estimated Time:2m 0s

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