Finding constants of an equation

[MathsHelpPlz]

"The height in metres of Jack's beanstalk t weeks after it was planted is given by the equation h=a((e^(ct)) - 1). Its heights at the ends of the first two weeks were 10cm and 50cm.
a) calculate the constants a and c."

I have done: h=a((e^(ct)) - 1)
So [1] 10=a(e^(ct)) - a
and 50=a(e^(2ct)) - a
=a((e^(2ct))-1)
[2] a=50/((e^(2ct))-1)

Substituting [2] into [1], giving 10=(50/((e^(2ct))-1))e^c - (50/((e^(2ct))-1))
then rearranging and turning it into a quadratic by leting e^c=x to get x=1 and x=4, giving c=Ln(4) as c cannot equal Ln(1).
But then when I then substitute c in to [1], i get a as 10/3, yet the book says the answer is 0.0333. I was wondering what I am doing wrong? Thank you for your time, any help will be greatly appreciated.

 

[lookagain]

"The height in metres of Jack's beanstalk t weeks after it was planted
is given by the equation h=a((e^(ct)) - 1).

Its heights at the ends of the first two weeks were 10cm and 50cm. **

> > > Do you mean its height at the end of the first week was 10 cm,
and that its height at the end of the second week was 50 cm? < < <

a) calculate the constants a and c."

I have done: h=a((e^(ct)) - 1)


So [1] 10=a(e^(ct)) - a
. . . The variable t is not supposed to be there, because you
substituted 1 for it (and 10 for h).


and 50=a(e^(2ct)) - a . . . See directly above.


=a((e^(2ct))-1)


[2] a=50/((e^(2ct))-1)

Substituting [2] into [1], giving 10=(50/((e^(2ct))-1))e^c - (50/((e^(2ct))-1))
then rearranging and turning it into a quadratic by leting e^c=x to get x=1 and x=4, giving c=Ln(4) as c cannot equal Ln(1).
But then when I then substitute c in to [1], i get a as 10/3, yet the book says the answer is 0.0333. I was wondering
what I am doing wrong? Thank you for your time, any help will be greatly appreciated.

I also worked out c = ln(4) and a = 10/3, but I still question the meaning (read: intent) of statement ** above.

 

[MathsHelpPlz]

I also worked out c = ln(4) and a = 10/3, but I still question the meaning (read: intent) of statement ** above.

Yea after the first week it was 10cm and then after the second it was 50cm. Ok thank you very much for your help, I thought I was going mad!

 

[HallsofIvy]

"The height in metres of Jack's beanstalk t weeks after it was planted is given by the equation h=a((e^(ct)) - 1). Its heights at the ends of the first two weeks were 10cm and 50cm.
a) calculate the constants a and c."

I have done: h=a((e^(ct)) - 1)
So [1] 10=a(e^(ct)) - a
and 50=a(e^(2ct)) - a
=a((e^(2ct))-1)
[2] a=50/((e^(2ct))-1)

Why do you still have "t" in there? You are told that its height 10 cm when t= 1 (after the first week) and 50 cm when t= 2 (after the second week) so you have \(\displaystyle 10= a(e^c- 1)\) and \(\displaystyle 50= a(e^(2c)- 1)\). I you simply divide the second equation by the first, you get \(\displaystyle 5= \frac{e^{2c}-1}{e^c- 1}\) so that \(\displaystyle 5e^{c}- 5= e^{2c}- 1\). That can be rewritten as \(\displaystyle e^{2c}- 5e^c+ 4= 0\). Let \(\displaystyle y= e^c\), and that becomes the quadratic equation \(\displaystyle y^2- 5y+ 4= 0\).

Substituting [2] into [1], giving 10=(50/((e^(2ct))-1))e^c - (50/((e^(2ct))-1))

then rearranging and turning it into a quadratic by leting e^c=x to get x=1 and x=4, giving c=Ln(4) as c cannot equal Ln(1).
But then when I then substitute c in to [1], i get a as 10/3, yet the book says the answer is 0.0333. I was wondering what I am doing wrong? Thank you for your time, any help will be greatly appreciated.[/QUOTE]

 

[MathsHelpPlz]

Why do you still have "t" in there? You are told that its height 10 cm when t= 1 (after the first week) and 50 cm when t= 2 (after the second week) so you have \(\displaystyle 10= a(e^c- 1)\) and \(\displaystyle 50= a(e^(2c)- 1)\). I you simply divide the second equation by the first, you get \(\displaystyle 5= \frac{e^{2c}-1}{e^c- 1}\) so that \(\displaystyle 5e^{c}- 5= e^{2c}- 1\). That can be rewritten as \(\displaystyle e^{2c}- 5e^c+ 4= 0\). Let \(\displaystyle y= e^c\), and that becomes the quadratic equation \(\displaystyle y^2- 5y+ 4= 0\).


then rearranging and turning it into a quadratic by leting e^c=x to get x=1 and x=4, giving c=Ln(4) as c cannot equal Ln(1).
But then when I then substitute c in to [1], i get a as 10/3, yet the book says the answer is 0.0333. I was wondering what I am doing wrong? Thank you for your time, any help will be greatly appreciated.

[/QUOTE]

Ah yes, thanks