Find the value of $a$.

After $4$ months $90$ fish were counted.

Find the value of $b$.

The number of fish in the pond will not decrease below $p$.

Write down the value of $p$.

$a{b^0} + 40 = 840$     (M1)

Note: Award (M1) for substituting $t = 0$ and equating to $840$.

$a = 800$     (A1)(C2)

$800{b^{ - 4}} + 40 = 90$     (M1)

Note: Award (M1) for correct substitution of their $a$ (from part (a)) and $4$ in the formula of the function and equating to $90$.

${b^4} = 16\;\;\;$OR$\;\;\;\frac{1}{{{b^4}}} = \frac{1}{{16}}\;\;\;$OR$\;\;\;b = \sqrt[4]{{16}}\;\;\;$OR$\;\;\;b = {16^{\frac{1}{4}}}$     (M1)

Notes: Award second (M1) for correctly rearranging their equation and eliminating the negative index (see above examples).

Accept $\frac{{800}}{{50}}$ in place of $16$.

OR

     (M1)(M1)

Notes: Award (M1) for a decreasing exponential and a horizontal line that are both in the first quadrant, and (M1) for their graphs intersecting.

For graphs drawn in both first and second quadrants award at most (M1)(M0).

$b = 2$     (A1)(ft)     (C3)

Note: Follow through from their answer to part (a) only if $a$ is positive.

$40$     (A1)     (C1)