Question

An evergreen nursery sells trees which grow at a rate of (dh)/(dt) = 2t + 4 where t is measured in years and his measured in feet. After growing for 2 years, a certain tree is 15 feet tall. How tall was the tree when it was planted ?

Answer 1

We can find the function h(t) by integrating the equation dh/dt, to get the following:

`\begin{gathered} h(t)=\int(dh)/(dt)dt=\int(2t+4)dt=t²+4t+C \\ \Rightarrow h(t)=t²+4t+C \end{gathered}`

we know that after growing for t = 2 years, a certain tree is 15 ft tall, this means that h(2) = 15, with this information we can find the value of C:

`\begin{gathered} h(2)=(2)²+4(2)+C=15 \\ \Rightarrow4+8+C=15 \\ \Rightarrow C=15-12=3 \\ C=3 \end{gathered}`

then, the equation to model the growth of the trees is h(t) = t²+4t+3

Finally, to find the height of the tree when it was planted, we can evaluate h(0) to find the initial height:

`\begin{gathered} t=0 \\ \Rightarrow h(0)=0²+4(0)+3=3 \\ h(0)=3 \end{gathered}`

therefore, the tree was 3 ft tall when it was planted