Understanding chemical kinetics
Is this the right formula to use to solve the above problem


Yes, this is it, if the initial concentrations of the reagents are the same (which is not mentioned explicitly in the problem statement, but probably is implied, because otherwise the problem would be unsolvable.)

Indeed, as A and B are spent in equal amounts, their concentrations are bound to be the same not only initially, but at all times: $[\ce{A}]=[\ce{B}]=c(t)$. Therefore $\dot c=-k\cdot[\ce{A}]\cdot[\ce{B}]=-k\cdot c^2$, which you seem to already know how to solve.

Update. Come to think of it, you don't actually need to solve for $c(t)$. You know that the rate is proportional to the first power of $[\ce{A}]$, and also to the first power of $[\ce{B}]$, and that both $[\ce{A}]$ and $[\ce{B}]$ have decreased by 50%; what would happen to the rate?

The concept of half life does not apply here, since the decay is not exponential.

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