# How do I work this out?

Wiremu planted a tree that was 1.5 m high.
He is told that the tree will increase in height at a rate of 8% a year.
The height h metres of the tree can be modelled by the function
h = 1.5(1 + 0.08)t
where t is the time in years since the tree was planted.
When will the height of the tree be 12 m?

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Integral
Staff Emeritus
Gold Member
Are you sure that you have the correct equation?

What have you done so far?

Are you sure that you have the correct equation?

What have you done so far?
WOW, what happened to my answer??? It took me 5 min to write that down, don't tell me that i waisted those 5 min..lol...

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Lol I don't have a clue how to start this question

You have an equation for height which is a function of time. You have the desired height and are looking for the required time. Can you rearrange the equation to find the time given the height?

I got the equation to rearange to t=1.62/h. Is that right?

Looks right to me.

I got the equation to rearange to t=1.62/h. Is that right?
The deal is that i do not think you have set right the very first equation at the first place. I wrote a detailed answer and posted it here, but i guess a moderator deleted it, due to forum rules, that we aren't supposed to give explicit answers.

Because i think that the differential equation that describes that situation, given that the height increases at a rate of 8% of the previous height is:

$$\frac{dh}{dt}=.08h$$ Now if you solve this diff. eq. you will end up with the solution function:

$$h(t)=1.5e^{.08t}$$

So, i think that this equation instead describes the height of the tree at any point in time.

Because i think that the differential equation that describes that situation, given that the height increases at a rate of 8% of the previous height is:

$$\frac{dh}{dt}=.08h$$ Now if you solve this diff. eq. you will end up with the solution function:

$$h(t)=1.5e^{.08t}$$

So, i think that this equation instead describes the height of the tree at any point in time.
It doesn't have anything with e. Now that I look carefully at the derivation of the thread author's formula, it's wrong. It does, however, have to deal with compounding.

The formula for compounding is F = I(1 + r)^t, not F = I(1 + r)t.

It doesn't have anything with e. Now that I look carefully at the derivation of the thread author's formula, it's wrong. It does, however, have to deal with compounding.

The formula for compounding is F = I(1 + r)^t, not F = I(1 + r)t.

Well, the deal is that this is an inappropriate way of describing the growth of a tree, since it implies that actually the tree grows only once a year in an abprupt way at a specific point in time, which is not true. The tree grows contnuously that is at every point in time. So, if we wanted to use the formula for compounding the appropriate one would be

$$F=I(1+\frac{r}{n})^{nt}$$ where I would be the initial value, n the number of compoundings per year, and t the number of years.

If we take $$n\rightarrow \infty$$ We actually get the result

$$F=Ae^{rt}$$ which is actually the solution to the differential equation

$$\frac{dF}{dt}=rF$$

So, in other words, the diff eq. that i used is the proper way to describe this situation.

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Integral
Staff Emeritus
Gold Member

I deleted your original post, because it was a complete solution.

Next, I saw no evidence in the origianal post that the Differential equation for a constant precentage growth rate was part of the problem. He was GIVEN an equation for the growth rate. The equation he posted does not meet the requirement of the given problem, therefore he has been asked to double check what he posted.

OP, please check what you have posted, I do not think your equation is correct, something is missing.

Here is an attachment to the exact question I was given... it is question 6

#### Attachments

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Here is an attachment to the exact question I was given... it is question 6
What is happening with this attachment??

The equation is h = 1.5(1 + 0.08)^t, not, h = 1.5(1 + 0.08)t

The equation is h = 1.5(1 + 0.08)^t, not, h = 1.5(1 + 0.08)t
Well, first you have to notice that this is a hypothetical situation, since like i explained before the tree grows continuously rather than just once a year, like it is claimed in this problem. So, lucky you, no need for diff. equations in this problem.

You are asked to find t-? when h(t)=12. In other words

$$12=1.5(1+0.08)^{t}$$ do you know how to isolate t here? jut take ln on both sides and you will be fine..lol...

How do you take the ln of both sides?

well, first devide throught my 1.5 and you'll get:
$$8=(1+0.08)^{t}$$ now we take the ln of both sides so we get

$$ln8=ln(1+0.08)^{t}$$, now do you know how to solve for t??

nope...

Well, we can see that the op has tried to solve this, but without sucess, so i think it is all right to go all the way through.

$$ln8=ln(1+0.08)^{t}$$

Or, better, i will just give some more hints, and try to put him/her on the right direction.

To the OP: Do you know the properties of logarithms? If not here are some:

$$log_a(xy)=log_ax+log_ay$$

$$log_ax^{n}=nlog_ax$$

$$log_a\frac{x}{y}=log_ax-log_ay$$ etc.

I hope this helps now. Does it? YOu need to figure it out which one of these rules to use on your problem. Remember that ln, is simply log base e. So, all these properties apply also at ln.

Not to bicker too much sutupidmath, but a continuous growth model is not correct. I'm looking out my window at this very moment. The view of all the snow and trees without leaves has served as a constant reminder for the last 6 months that tree growth isn't a continuous growth model. There's absolutely NOTHING wrong with the OP's equation, even in cases of a continuous growth model, providing t is measured in whole years. The original problem states that it's 8% "annual" growth. i.e. at the end of 1 year, it's 8% taller than at the beginning of the year. Thus, if you were going to switch to a continuous growth model, you would have to solve for r; it wouldn't be .08. And, while differential equations could be used, assuming continuous growth, they're not necessary to solve this problem. (Not to mention that this is the pre-calculus section. Last I checked, diff eq's weren't a pre-requisite for pre-calc.)

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Not to bicker too much sutupidmath, but a continuous growth model is not correct. I'm looking out my window at this very moment. The view of all the snow and trees without leaves has served as a constant reminder for the last 6 months that tree growth isn't a continuous growth model. There's absolutely NOTHING wrong with the OP's equation, even in cases of a continuous growth model, providing t is measured in whole years. The original problem states that it's 8% "annual" growth. i.e. at the end of 1 year, it's 8% taller than at the beginning of the year. Thus, if you were going to switch to a continuous growth model, you would have to solve for r; it wouldn't be .08. And, while differential equations could be used, assuming continuous growth, they're not necessary to solve this problem.
Well, i don't want to blindly argue that a continuous model is actually correct, because since it is claimed that the tree was just planted, the fact that we cannot notice the growth of the tree at every moment in time, doesn't necessarly mean that the tree in fact does not grow continuously.

HOwever, since the question actually is estimating the tree growth only once a year, that formula that the op presented actually can be used, like you said.

(Not to mention that this is the pre-calculus section. Last I checked, diff eq's weren't a pre-requisite for pre-calc.)
LOL, what was the last time you checked? It has become a standard prerequisite of pre-calc nowa days!!

Sorry, this is a bad joke!

does the eqation go to: ln8=t*ln+0.08?

And what is the difference between log and ln?

does the eqation go to: ln8=t*ln+0.08?

And what is the difference between log and ln?