# Differential equation tree height

1. Mar 8, 2016

### chwala

1. The problem statement, all variables and given/known data
A tree is planted as a seedling of negligible height. The rate of increase of its height , in metres per year is given by $0.2√(25-h)$
a. explain why tree cant exceed 25 metres. answer⇒
$dh/dt=0$ when h=25
b. express t as a function of h answer⇒ $t=-10√(25-h)+50$
c. how long does it take for tree to put on (i) its first metre (ii)its last metre
d. express h as a function of t here i did it like this $h=25-(t^2-100t+2500/100)$
$h=2500-t^2+100t-2500/100$
$h= -t^2+100t/100$
$h=t-0.01t^2$ which agrees with textbook answer. where $0≤t≤50$

2. Relevant equations

3. The attempt at a solution
c(i)$t=-10√25-1+50 t=1.0$ years , this is the correct answer as per text book.
c(ii) solution is $t=10$ years implying that h=9
my question is why $h=9$?why not 15 or 20? i was unable to solve c(ii)

Last edited: Mar 8, 2016
2. Mar 8, 2016

### Qwertywerty

Actually, Δt = 10 years. It doesn't imply that h = 9.
Hint : You simply have to find the time taken for change from h = 24m to h = 25m.

Hope this helps.

3. Mar 8, 2016

### HallsofIvy

Staff Emeritus
I think you are misunderstanding (c), either the question or the answer. (cii) asks how long it will take the tree to grow its last meter. You have correctly shown that the tree will grow to 25 m so its "last meter" will be from 24 to 25 meters. It take $t= 50- 10\sqrt{25- 24}= 50- 10= 40$ years to grow to 24 m and $t= 50- 10\sqrt{25- 25} = 50$ years to grow to 25 m. It takes 50- 40= 10 years to grow that last 1 m. The solution is NOT "t= 10" because t is the number of years the tree has been growing, NOT the difference in years. h= 9 m is the tree's height when t= 10. That has nothing to do with (cii).

4. Mar 8, 2016

### chwala

Thanks its now clear to me, the language used in questions is sometimes confusing. greetings from Africa.