 # 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 Kaycee Roche 2020-11-24 Answered
Below is data collected from the growth of two different trees over time. Each tree was planted in 1960, and the trees
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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 ​ .