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The correct answer is $.09$. It's given that the expression $0.36x^2+0.63x+1.17$ can be rewritten as $a\left(4x^2+7x+13\right)$. Applying the distributive property to the expression $a\left(4x^2+7x+13\right)$ yields $4a{x}^2+7ax+13a$. Therefore, $0.36x^2+0.63x+1.17$ can be rewritten as $4a{x}^2+7ax+13a$. It follows that in the expressions $0.36x^2+0.63x+1.17$ and $4a{x}^2+7ax+13a$, the coefficients of $x^2$ are equivalent, the coefficients of $x$ are equivalent, and the constant terms are equivalent. Therefore, $0.36=4a$, $0.63=7a$, and $1.17=13a$. Solving any of these equations for $a$ yields the value of $a$. Dividing both sides of the equation $0.36=4a$ by $4$ yields $0.09=a$. Therefore, the value of $a$ is $0.09$. Note that .09 and 9/100 are examples of ways to enter a correct answer.