Question

the table below shows the population size of tasmanian wolf $t$ years after 1900. graph the exponential function that respresents the situation, rounding to the nearest integer whenever necessary.

year

population size of tasmanian wolf

----

----

1900

66

1901

55

1902

47

1903

39

1904

33

show your work here
draw

Explanation:

Step1: Define exponential decay model

The general exponential decay function is $P(t) = P_0 \cdot r^t$, where $P_0$ is the initial population, $r$ is the decay rate, and $t$ is years after 1900.

Step2: Identify initial population

In 1900 ($t=0$), $P_0 = 66$.

Step3: Calculate decay rate $r$

Use 1901 data ($t=1$, $P=55$):
$r = \frac{P(1)}{P_0} = \frac{55}{66} \approx 0.833$
Verify with 1902: $66 \cdot (0.833)^2 \approx 66 \cdot 0.694 \approx 46.8 \approx 47$, which matches the table.

Step4: Finalize the function

$P(t) = 66 \cdot (0.833)^t$

Step5: List plot points

Calculate (or use given) points:

• $t=0$: $P=66$
• $t=1$: $P=55$
• $t=2$: $P=47$
• $t=3$: $P=39$
• $t=4$: $P=33$

Answer:

The exponential function is $P(t) = 66 \cdot (0.833)^t$. To graph:

1. Label the x-axis "Years after 1900 ($t$)" and y-axis "Population Size".
2. Plot the points $(0, 66)$, $(1, 55)$, $(2, 47)$, $(3, 39)$, $(4, 33)$.
3. Draw a smooth curve through the points, trending downward as $t$ increases.