1. Another problem

Here's the last of the problems that i'm not sure how to tackle.
A group of students where studying how fast a plant was growing. Once a day they measured the plant. These was the results:
Day--- length (cm)
0 -----0,5
1 -----0,8
2 -----1,3
3 -----2,1
4 -----3,3
5 -----5,2
6 -----8,1
7 -----13,4

The students created an equation from this information: y=0,5*1,6^X
X is time in days. Y is the height of the plant.
How did the students work out 1,6 from this information?

2. Hello, AknightwhosayNi!

A group of students where studying how fast a plant was growing.
Once a day they measured the plant. These are the results:

. . $\displaystyle \begin{array}{c|c} \text{Day} & \text{Height} \\ \hline 0 & 0.5 \\ 1 & 0.8 \\ 2 & 1.3 \\ 3 & 2.1 \\ 4 & 3.3 \\ 5 & 5.2 \\ 6 & 8.1 \\ 7 & 13.4 \end{array}$

The students created an equation from this information: .$\displaystyle y\:=\:0.5\cdot 1.6^x$
. . where $\displaystyle x$ is time in days, $\displaystyle y$ is the height of the plant in cm.

How did the students work out 1.6 from this information?

They conjectured that the function is of the form: .$\displaystyle y \;=\;A\cdot B^x$

To determine $\displaystyle A$ and $\displaystyle B$, use two values from the table.
. . We'll use the first two values . . .

$\displaystyle x=0,\;y=0.5\!:\quad 0.5 \:=\:A\cdot B^0 \quad\Rightarrow\quad A \:=\:0.5$

. . Hence, the function (so far) is: .$\displaystyle y \;=\;0.5\cdot B^x$

$\displaystyle x = 1,\;y = 0.8\!:\quad 0.8 \:=\:0.5B^1 \quad\Rightarrow\quad B \:=\:\frac{0.8}{0.5} \:=\:1.6$

Therefore, the function is: .$\displaystyle y \;=\;0.5\cdot1.6^x$

3. I see... Thanks!