Question

In 2009, a tree was 10 feet tall. In 2012, the tree was 16 feet tall. Construct a linear model that represents the height, h, of the tree t years after 2006.

Answer 1

h(t)=2t+4

Explanation:

In 2009, h(t=0)=10 feet

In 2012, h(t=3)=16 feet

Let us determine the rate of the growth of the tree.

[TeX]m=\frac{h_{2}-h_{1}}{t_{2}-t_{1}}[/TeX]

[TeX]m=\frac{16-10}{3-0}[/TeX]

[TeX]m=\frac{6}{3}=2[/TeX]

The equation of the trees growth can be written as:

h(t)=2t+c

When h=16, t=3

Using these values to solve for c

16=2(3)+c

c=16-6=10

Therefore the equation of the tree's growth is:

h(t)=2t+10

To determine the linear model that represents the tree's growth after 2006, i.e. three years before 2009

We substitute t=t-3

h(t)=2(t-3)+10

=2t-6+10=2t+4

The linear model that represents the tree's growth t years after 2006 is:

h(t)=2t+4

Answer 2

h = 2t + 4

Explanation:

Solution:-

- The height of the tree = h

- The number of years elapsed after 2006 = t

- height h1 = 10 ft, t1 = ( 2009 - 2006 ) = 3 yrs

- height h2 = 16 ft, t2 = ( 2012 - 2006 ) = 6 years

- The linear model of tree height "h" as a function of time "t" years after 2006 can be expressed in the form:

h = m*t + c

Where, m = The rate of change of height

c = The initial height of tree in 2006.

- We will use the given data to evaluate constant "m".

Rate = m = ( h2 - h1 ) / ( t2 - t1)

m = ( 16 - 10 ) / ( 6 - 3 )

m = 6 / 3 = 2

- Then use "m" and given set of data to evaluate the initial height of tree "c" in year 2006.

h = 2*t + c

h1 = 2*t1 + c

c = 10 - 2*3 = 4 ft

- The linear model for the height of tree "h" as fucntion of time "t" years after 2006 will be:

h = 2t + 4