Answer: A. $0.13$

Explanation:

$1$ . Write out the two given conditions using the function definition:

   - $f(s) = ab^s = t$

   - $f(s+1) = ab^{s+1} = t - 0.87t = 0.13t$

$2$ . Use the two equations to find the value of b:

   - The constant b in an exponential function represents the growth/decay factor, which is the ratio of consecutive terms, $\dfrac{f(x+1)}{f(x)}$ .

   - Let's find this ratio using the given information:

     $\dfrac{f(s+1)}{f(s)} = \dfrac{0.13t}{t}$

$3$ . Simplify the ratio:

   - On the right side, the 't's cancel out: $\dfrac{f(s+1)}{f(s)} = 0.13$ .

   - Now let's look at the left side using the function definition:

     $\dfrac{ab^{s+1}}{ab^s}$

   - The 'a's cancel out. Using exponent rules, $b^{s+1} / b^s = b^{(s+1)-s} = b^1 = b$ .

   - So, the ratio $\dfrac{f(s+1)}{f(s)}$ is equal to b.

$4$ . Equate the results:

   - We found that the ratio is equal to b and is also equal to $0.13$ .

   - Therefore, $b = 0.13$ .

$5$ . Conclusion:

   - The value of b is $0.13$ , which corresponds to option A.