The height in metres of Jack's beanstalk $t$ weeks after it was planted is given by $$h = a(e^{ct} -1)$$
Its heights at the end of the first two weeks were $10cm$ and $50cm$.
Calculate the constants $a$ and $c$
I can't seem to get the correct answer using simultaneous equations. How should I be going about this?
Thanks in advance
When you substitute the two constraints in, we get $$ 0.5 = a \left(e^{2c}-1\right)\\ 0.1 = a \left(e^c-1\right). $$ It may be easier to see the next step if we give $e^c$ some name, say $d$; with this choice of parameters, the equations above become $$ 0.5 = a \left(d^2 - 1 \right)\\ 0.1 = a \left(d-1\right). $$ Factoring the first equation gives the pair $$ \begin{align} 0.5 &= a\left(d-1\right)\left(d+1\right)\\ 0.1 &= a\left(d-1\right), \end{align} $$ from which it should be straightforward to find $d$, followed by $a$ and $c$.
(It's not actually necessary to replace $e^c$ with $d$, but it helps motivate factoring $e^{2c}-1$ into $\left(e^c-1\right)\left(e^c+1\right)$.)