Question

c. assuming that growth is exponential, complete the table.
\td. write an exponential model for ( z(t) ).

( t )	0	1	2	3	4

Explanation:

Step1: Identify initial value

At $t=0$, $z(0)=5$, so $a=5$ (exponential form $z(t)=a \cdot b^t$).

Step2: Find growth factor $b$

Use $t=2$, $z(2)=45$:
$$45 = 5 \cdot b^2$$
$$b^2 = \frac{45}{5}=9$$
$$b=3 \quad (\text{since growth is positive})$$

Step3: Fill table values

• $t=1$: $z(1)=5 \cdot 3^1=15$
• $t=3$: $z(3)=5 \cdot 3^3=5 \cdot 27=135$
• $t=4$: $z(4)=5 \cdot 3^4=5 \cdot 81=405$

Step4: Write exponential model

Substitute $a=5$, $b=3$ into $z(t)=a \cdot b^t$.

Answer:

Completed Table:

$t$	$z(t)$
1	15
2	45
3	135
4	405

Exponential Model:

$z(t)=5 \cdot 3^t$