Answer: D. $69$

Explanation:1. Understand the problem: The function is $f(x) = 24(1.30)^{\dfrac{x}{3}}$. The value of $f(x)$ increases by $p\%$ for every increase of $x$ by $6$. We need to find the value of $p$.

2. Evaluate the function at $x$ and $x+6$: Let $f(x\_1) = 24(1.30)^{\dfrac{x\_1}{3}}$. Let $f(x\_1+6) = 24(1.30)^{\dfrac{x\_1+6}{3}}$.

3. Find the ratio $\dfrac{f(x\_1+6)}{f(x\_1)}$: $\dfrac{f(x\_1+6)}{f(x\_1)} = \dfrac{24(1.30)^{\dfrac{x\_1+6}{3}}}{24(1.30)^{\dfrac{x\_1}{3}}}$ $= (1.30)^{\dfrac{x\_1+6}{3} - \dfrac{x\_1}{3}}$ $= (1.30)^{\dfrac{x\_1+6-x\_1}{3}}$ $= (1.30)^{\dfrac{6}{3}}$ $= (1.30)^2$

4. Calculate $(1.30)^2$: $(1.30)^2 = 1.3 \times 1.3 = 1.69$. So, $f(x\_1+6) = 1.69 \times f(x\_1)$.

5. Determine the percentage increase $p$: If the value increases to $1.69$ times its original value, the increase factor is $1.69$. The percentage increase is $(1.69 - 1) \times 100\%$. $(0.69) \times 100\% = 69\%$. So, $p = 69$.

6. Compare with the given options: [A] $30$ [B] $41$ [C] $60$ [D] $69$ Our calculated value for $p$ is $69$, which matches option [D].

7. Conclusion: The value of $p$ is $69$.