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**Hint:**We know that Rate is related to the speed. The rate of the reaction determines the speed of the reaction, that is how fast or slow is the reaction. The branch of physical chemistry which determines the rate of the reaction is known as chemical kinetics.

**Complete answer:**

The rate law in chemical kinetics can be defined as a mathematical relationship obtained by experimenting and comparing the rate of the reaction to the concentration of the reactant. The order of the reaction is basically the sum of the powers to which the concentration term of the reactants is raised to. The rate law expression is \[R=k{{\left[ A \right]}^{m}}{{\left[ B \right]}^{n~}}......\left( 1 \right)\]

The rate of formation of C increases by a factor of \[2.82\] on doubling the concentration of A. The rate law expression becomes \[R'=2.82R=k{{2}^{m}}{{\left[ A \right]}^{m}}{{\left[ B \right]}^{n}}~......\left( 2 \right)\]

Divide equation \[\left( 2 \right)\] with equation \[\left( 1 \right)\]

\[\dfrac{2.82R}{R}=\dfrac{k{{2}^{m}}{{\left[ A \right]}^{m}}{{\left[ B \right]}^{n}}}{k{{\left[ A \right]}^{m}}{{\left[ B \right]}^{n}}}.\text{ }\Rightarrow 2.82=2m\text{ }\]

Thus we get; \[m=\dfrac{3}{2}.\]

The rate of formation of C increases by a factor of \[9\] on tripling the concentration of B. The rate law expression becomes \[R''=9R=k{{\left[ A \right]}^{m}}{{3}^{n}}{{\left[ B \right]}^{n}}\text{ }......\left( 3 \right)\]

Divide equation \[\left( 3 \right)\] with equation \[\left( 1 \right)\]

\[\dfrac{9R}{R}=\dfrac{k{{\left[ A \right]}^{m}}{{3}^{n}}{{\left[ B \right]}^{n}}}{\text{ }k{{\left[ A \right]}^{m}}{{\left[ B \right]}^{n}}}\Rightarrow 9={{3}^{n}}\]

Thus, \[n=2.\]

The overall order of the reaction is \[m+n=\dfrac{3}{2}+2=\dfrac{7}{2}.\]

**Therefore, the correct answer is option A.**

**Note:**

Remember the rate of the reaction is important to determine the speed of the reaction. The unit of the rate of the reaction is the concentration per unit time. If the concentration is in molar and time is in seconds then the unit of the rate is molar per second.

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