Question

anted, a tree grows at a rate of 0.6 meters per year. After 3 years the tree is 2.2 meters tall. Write then point-slope form that models the situation. Then, predict the height of the tree after 6 years.

Answer 1

- The question tells us a tree grows at the rate of 0.6 meters per year. In comparison to the equation of a line, the slope (m) is the rate at which x changes with respect to y. Thus, we can conclude that the slope of the growth of the tree is 0.6 meters per year.

- This implies that y must be the height of the tree while x is the number of years that have passed.

- Thus, we can write the equation of the growth of the tree as:

`\begin{gathered} y=mx+c \\ \text{where,} \\ y=\text{Height of the tree after x years} \\ x=\text{ Number of years} \\ m=\text{The rate at which the tree } \\ c=\text{Initial height of the tree before the first year.} \end{gathered}`

- We need to find the value of c. This can be gotten if we simply substitute y = 2.2 while x = 3 since the question says after 3 years, the tree grows 2.2 meters. We have already established that m = 0.6, thus, we can find c as follows:

`\begin{gathered} y=mx+c \\ y=2.2,x=3,m=0.6 \\ \\ 2.2=0.6(3)+c \\ 2.2=1.8+c \\ \text{Subtract 1.8 from both sides} \\ c=2.2-1.8 \\ c=0.4 \end{gathered}`

- Thus, we can write the equation as follows:

`\begin{gathered} y=mx+c \\ y=0.6x+0.4 \\ \\ \text{This can be rewritten as:} \\ y-2.2=0.6(x-3) \end{gathered}`

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- In order to find the height of the tree after 6 years, we simply substitute x = 6 into our newly formed equation. This is done below:

`\begin{gathered} y=0.6x+0.4 \\ \text{put }x=6 \\ \\ \therefore y=0.6(6)+0.4 \\ y=3.6+0.4 \\ \therefore y=4 \end{gathered}`

- Thus, the height of the tree after 6 years is 4 meters

Final Answer

- The equation of the growth of the tree is:

`y-2.2=0.6(x-3)`

- The height of the tree after 6 years is 4 meters