Step 1 :The problem is asking when the height of the fireball, modeled by the function \(H(t)=-10 t^{2}+50 t+6\), will be equal to the height of the laser, modeled by the function \(L(t)=7.9 t-10.6\). This means we need to find the time \(t\) when \(H(t) = L(t)\).
Step 2 :This can be solved by setting the two equations equal to each other and solving for \(t\).
Step 3 :The solution to the equation gives two values for \(t\). However, since time cannot be negative in this context, we discard the negative value.
Step 4 :Therefore, the laser will explode the fireball at approximately 4.57 seconds.
Step 5 :Final Answer: The laser will explode the fireball at approximately \(\boxed{4.57}\) seconds.