Answer: A. $y=70(1.50)^{x}$

Explanation:1. Understand the general form of an exponential equation: An exponential equation relating y and x is typically of the form $y = ab^x$, where 'a' is the initial value (value of y when $x=0$) and 'b' is the growth factor per unit increase in x.

2. Determine the initial value 'a': The problem states that when $x=0$, the value of $y$ is $70$. Substituting $x=0$ into $y=ab^x$: $y = ab^0 = a(1) = a$. So, the initial value $a = 70$.

3. Determine the growth factor 'b': "For every increase in the value of x by $1$, the corresponding value of y increases by $50\%$ of its previous value." If the value of y increases by $50\%$, the new value is the original value ($100\%$) plus the increase ($50\%$), which is $100\% + 50\% = 150\%$ of the previous value. Converting $150\%$ to a decimal, the growth factor $b = \dfrac{150}{100} = 1.50$.

4. Write the exponential equation: Substitute $a=70$ and $b=1.50$ into the form $y = ab^x$: $y = 70(1.50)^x$

5. Check the solution with the options: The derived equation is $y=70(1.50)^{x}$. This matches option [A].

6. Conclusion: The equation that represents this relationship is $y=70(1.50)^{x}$.