Question

The height of a tree increases by approximately 2% each year. The tree is currently 45 feet tall. Which expression will calculate how tall the tree will be in 1 year?

Answer 1

Answer:
`h=45e^{(t)/(50) }` feet

Explanation:

Let
`h` be the height of the tree at time
`t`.

Given that the height of a tree increases by approximately
`2`% each year.

So,
`(dh)/(dt)=(2)/(100)h`

`(dh)/(h)=(2)/(100)dt`

Given that when
`t=0`,
`h=45 feet`

Integrating on both sides,

`\int\limits^h_(45) {(dh)/(h) } =\int\limits^t_0 {(2)/(100) } \, dt`

`ln(h)-ln(45)=(2)/(100)(t-0)`

`ln(h)=(2)/(100)t+ln(45)`

applying anti-logarithm on both sides,

`h=45* e^{(t)/(50)}`