A ball is rolling in a straight line such that its distance away from the starting point, $s$ cm, can be modelled using the equation
$s = at + \frac{b}{\sqrt{t} + 4} + c,$
where $t$ is the time taken in seconds, and $a$, $b$ and $c$ are real constants. The ball is at the starting point when $t = 0$, and moved 10 cm in the first 5 seconds. It moved another 9 cm in the next 16 seconds. Find the ball's distance away from the starting point when $t = 50$.

1. Use the initial condition at $t = 0$ to express $c$ in terms of $b$.

When $t = 0$, the ball is at the starting point, meaning $s = 0$.
$0 = a(0) + \frac{b}{\sqrt{0} + 4} + c$
$0 = 0 + 0.25b + c \implies c = -0.25b$

2. Use the condition for the first 5 seconds to form an equation relating $a$ and $b$.

When $t = 5$, the distance $s = 10$.
$10 = 5a + \frac{b}{\sqrt{5} + 4} + c$
Substituting $c = -0.25b$:
$10 = 5a + b\left(\frac{1}{\sqrt{5} + 4} - 0.25\right)$

3. Use the condition for the next 16 seconds to form a second equation relating $a$ and $b$.

The ball moved another $9$ cm in the next $16$ seconds, so at $t = 5 + 16 = 21$, the distance from the start is $s = 10 + 9 = 19$.
$19 = 21a + \frac{b}{\sqrt{21} + 4} + c$
Substituting $c = -0.25b$:
$19 = 21a + b\left(\frac{1}{\sqrt{21} + 4} - 0.25\right)$

4. Solve the simultaneous equations to find the values of $a$, $b$, and $c$.

Equating (1) and (2):
$2 + \frac{\sqrt{5}}{20(\sqrt{5} + 4)}b = \frac{19}{21} + \frac{\sqrt{21}}{84(\sqrt{21} + 4)}b$
$b \left( \frac{\sqrt{5}}{20(\sqrt{5} + 4)} - \frac{\sqrt{21}}{84(\sqrt{21} + 4)} \right) = \frac{19}{21} - 2$
Using calculator values to at least 5 significant figures for intermediate accuracy:
$b (0.017928 - 0.0063564) \approx -1.0952$
$0.011572 b \approx -1.0952 \implies b \approx -94.645$
Substitute $b$ back to find $a$ and $c$:
$c = -0.25(-94.645) \approx 23.661$
$a \approx 2 + 0.017928(-94.645) \approx 0.30316$

5. Calculate the distance $s$ when $t = 50$.

Substitute $a$, $b$, $c$ and $t = 50$ into the model:
$s = 50(0.30316) + \frac{-94.645}{\sqrt{50} + 4} + 23.661$
$s \approx 15.158 - 8.549 + 23.661 = 30.270$
Rounding to 3 significant figures, $s \approx 30.3$ cm.