# A mathematical model has an equation y = ae^(bx) and this curve passes through the points  A(0,1/2) and B(4,5) . Find a and b?

Sep 18, 2017

$a = \frac{1}{2}$ and $\frac{1}{4} \ln 10$

With these results we have:

$y = \frac{1}{2} {e}^{\left(\frac{1}{4} \ln 10\right) x}$

#### Explanation:

We have:

$y = a {e}^{b x}$

as a model. And we have two data points:

$A \left(0 , \frac{1}{2}\right)$ and $B \left(4 , 5\right)$

Using $A$ we have:

$\frac{1}{2} = a {e}^{0} \to a = \frac{1}{2}$

Using $B$ we have:

$5 = \frac{1}{2} {e}^{b 4} \implies {e}^{4 b} = 10$
$\therefore 4 b = \ln 10$
$\therefore b = \frac{1}{4} \ln 10$

With these results we have:

$y = \frac{1}{2} {e}^{\left(\frac{1}{4} \ln 10\right) x}$

Which we can see graphically: