Answer
a) $
P=P_0-0.1P_0=0.9P_0
$
b) $
P=P_0(0.9)^t
$
Work Step by Step
A) Given that $P_0$ is the initial population and that it decreases by $10\%$ per year, the balance population after one year is
$$
P=P_0-0.1P_0=0.9P_0
$$
After two years, the decrease is
$$
P=0.9P_0-0.1P_0=0.8P_0
$$
This continues until it is zero at the end of the 10 years. A general formula of the population can be expressed as
$$
P=P_0-0.1P_0 t
$$
See the graph below.
B) If the population decreases exponentially, the population after one year is
$$
P=P_0-0.1P_0=0.9P_0
$$
After two years, the decrease is
$$
P=0.9 P_0-0.1\left(0.9 P_0\right)=0.9 P_0(1-0.1)=P_0(0.9)^2
$$
A general formula of the population can be expressed as
$$
P=P_0(0.9)^t
$$
Unlike the linear case, the population after 10 years is
$$
P=P_0(0.9)^{10}=0.35 P_0=35 \% P_0
$$
The population decreased to $35 \%$ at the end of 10 years. A plot of both functions is shown in the figure below for an initial population of $P_0 = 50$.
