Solving Word Problem Logs with Population of Deer Increase from 800 to 1800

[Taylor_1989]

I have attached a print screen of this question. Now I have solved the problem, but only with the help of the marks, scheme. Don't get me wrong, I didn't just look at the MS and assumed it was the right ans. I kind of of reverse engine the ans.

My problem is with part b) It tells me to use the model to predict the number of years needed for the population of deer to increase from 800 to 1800.

I highlighted predict for a reason, which I will explain in a bit. Now from the previous question I know that $a=1.1776$

so If I solve the equation in terms of a^t I get this $a^t=36$ now if I insert the value of a into this a take log_1.177636=t, I get a value close to 22; I don't have the exact vale to hand, so I assumed when the question predict it wants me to round to the nearest whole number.

Now my issue is that originally I sub the actual value of a into the original equation and obviously got the wrong ans. But my question is why do I have to wait until the last part of solving to insert the value of a. The only conclusion I have managed to come to is that because the question states the a is a constant the value will not change so I should not multiple through in the original equation because it would change the value of the constant.

I would appreciate any sort of advice on how to do these types of questions, I have been looking at the definition of "constant" in math terms which is why I drew the conclusion of what I have said.

Big thanks in advance

p.s the sub script of "a" in the print screen equation is t.

 

[SteamKing]

You are given an initial population and the time it took to obtain that population.

This data is used to find the constant a. In answering the question about the growth in population after that initial time, you solve an equation with one unknown instead of two.

BTW, the answer to (b) is not 22.

 

[Taylor_1989]

What do you mean the ans is not 22? I took the log on an online cal, which gave me something like 21.8... Something like that, which I rounded to 22 due to the fact it said predict. The MS say it 22 as well, but I have see errors on ms like this before. I am right in take the log, to get T? Also, when you say about solving the equation with unknown instead of two, why can't I multiple the formal give by the value of a and then just simplify and solve for T?

 

[Taylor_1989]

Okay been researching trying get to grip with these sort of problems. So the 1.117.. Of the equation is what you would call a growth constant? Why can't I time the new appointed growth constant by 2000 ect in the formula given? I would know not when doing a question of this sort but, would like why?

 

[SammyS]

What do you mean the ans is not 22? I took the log on an online cal, which gave me something like 21.8... Something like that, which I rounded to 22 due to the fact it said predict. The MS say it 22 as well, but I have see errors on ms like this before. I am right in take the log, to get T? Also, when you say about solving the equation with unknown instead of two, why can't I multiple the formal give by the value of a and then just simplify and solve for T?

You may well get t = 21.92... ≈ 22 years.

That's the time for the population to go from what number of deer to 1800 deer ?

Do you know the initial population?

 

[Taylor_1989]

Okay I think I am catching on, I don't know the initial population so I would need to take how long it took for the initial population and I.e 22 years and take from how long it took to get the 800 population so, 6 years. So 22-6=16?

 

[SammyS]

Okay I think I am catching on, I don't know the initial population so I would need to take how long it took for the initial population and I.e 22 years and take from how long it took to get the 800 population so, 6 years. So 22-6=16?

Right.

 

[Taylor_1989]

I am still having trouble with a though, what is it exactly? Is it a growth constant and if so, surly it should be a whole number as, you can't get a decimal point of deer?

 

[SammyS]

I am still having trouble with a though, what is it exactly? Is it a growth constant and if so, surly it should be a whole number as, you can't get a decimal point of deer?

What is what exactly? To what are you referring ?

 

[Taylor_1989]

Sorry did not put in bold. I am having trouble with the "a" part of the formula. It says a is a constant term, from what I read it is a growth constant, but how it that possible when the growth is a decimal. And why when I have to find "t"can't I just multiply the equation but the growth constant, and then solve for t.

 

[SammyS]

Sorry did not put in bold. I am having trouble with the "a" part of the formula. It says a is a constant term, from what I read it is a growth constant, but how it that possible when the growth is a decimal. And why when I have to find "t"can't I just multiply the equation but the growth constant, and then solve for t.

I'm not sure what all you mean, but I'll give a try at an answer.

This growth model is not exponential growth. The parameter a is indeed a constant. It has a decimal representation, but this representation is as an infinite non-repeating decimal. You just found an approximate value. In fact, the exact value of a in this problem is given by:

$\displaystyle a=\sqrt[6]{\frac{8}{3}}$​

(That's the sixth root of 8/3 .)

 

[iRaid]

SORRY wrong post.

 

[Taylor_1989]

Will try explain a bit better: so in the part b. of the question I am ask to find the value of t.
What I am asking is this : $p=\frac{2000(1.1778)^t}{4+1.1178^t}$. Why can't I multiple 2000 by 1.1178 and 4+1.1778^t(1800) " after cross multiplying the latter" but I think I have ans my own question a won't be able to multiple $[2000*1.1778^t]$ due to the different base rule? Correct so I would leave a in the formula because it just easier to simplify.

 

[SteamKing]

You don't have to complicate things by bringing different bases into play.

If a^t = 36, the one can take logs of both sides, so that:

t log a = log 36

t = log 36 / log a