# Below is data collected from the growth of two different trees over time. Each tree was planted in 1960, and the tree's height has been collected ever

Below is data collected from the growth of two different trees over time. Each tree was planted in 1960, and the trees
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

joshyoung05M

Tree A is curved which means it could be an exponential function. The yy-coordinates are 10, 20, 40, and 80. Since the yy-coordinates are doubling every 10 years, then the function is exponential since exponential functions have a constant factor (the number you multiply by).
2. Unlike the yy-coordinates for Tree A, the yy-coordinates for Tree B are not increasing by a constant factor. Since $37-12=25,62-37=25$, and $87-62=25$, then the yy-coordinates for Tree B are increasing by a constant amount of 25. This means the function has a constant rate of change of 25 feet per 10 years. The function is then linear since linear functions have constant rates of change.
Exponential functions are of the form $y=a\cdot {b}^{x/c}$ where aa is the amount when $x=0$, b is the growth factor, and c is how often the amount changes by the growth factor. Since the y-coordinate is 10 when $x=0$ (the year 1960), then $a=10$. Since the y-coordinates are doubling every 10 years, then . The function for Tree A is then $y=10\cdot {2}^{x/10}$.
Linear functions are of the form $y=mx+b$ where mm is the constant rate of change and bb is the y-intercept. Since the rate of change is 25 feet per 10 years, then $m=25/10=2.5m$. Since the height is 12 feet when $x=0$ (the year 1960), then $b=12$. The function for Tree B is then $y=2.5x+12.$
From the previous problems, Tree A is growing by a constant factor of 2 every 10 years and Tree B is growing by a constant amount of 25 feet every 10 years. In the long term, Tree A has a greater growth rate than Tree B since growing by a constant factor will give greater increases than growing by a constant amount as time increases.
5. From the graph, the initial height of Tree A was 10 feet in 1960. From the table, the initial height of Tree B was 12 feet in 1960. Therefore, Tree B had a greater initial height.
6. From the graph and table, in 1990 Tree A had a height of 80 feet and Tree B had a height of 87 feet. Tree A's height will then exceed Tree B's height sometime after 1990. Since 1990 is 30 years after 1960, then $x=30x=30$ corresponds to 1990. Make a table finding the heights of the trees after 1990:
The height of Tree A will then exceed the height of Tree B after $x=33x=33$ years which corresponds to 1993 1993 ​ .