SAT Question e1391dd6: Nonlinear Functions answer and explanation with Desmos solution

Choice B is correct. Let $x$ be the number of years after $1985$. It follows that $\frac{x}{2}$ represents the number of $2$-year periods that will occur within an $x$-year period. According to Moore's law, every $2$ years, the number of transistors included on microprocessors is estimated to double. Therefore, $x$ years after $1985$, the number of transistors will double $\frac{x}{2}$ times. Since the number of transistors included on a microprocessor was $275,000$, or $0.275$ million, in $1985$, the estimated number of transistors, in millions, included $x$ years after $1985$ can be modeled as $0.275\times 2^{\frac{x}{2}}$. The year in which the number of transistors is estimated to be $1.1$ million is represented by the value of $x$ when $1.1=0.275\times 2^{\frac{x}{2}}$. Dividing both sides of this equation by $0.275$ yields $4=2^{\frac{x}{2}}$, which can be rewritten as $2^2=2^{\frac{x}{2}}$. Since the exponential equation has equal bases on each side, it follows that the exponents must also be equal: $2=\frac{x}{2}$. Multiplying both sides of the equation $2=\frac{x}{2}$ by $2$ yields $x=4$. Therefore, according to Moore's law, $4$ years after $1985$, or in $1989$, the number of transistors included on microprocessors is estimated to reach $1.1$ million.

Alternate approach: According to Moore's law, $2$ years after $1985$ (in $1987$), the number of transistors included on a microprocessor is estimated to be $2\times 275,000$, or $550,000$, and $2$ years after $1987$ (in $1989$), the number of transistors included on microprocessors is estimated to be $2\times 550,000$, or $1,100,000$. Therefore, the year that Moore's law estimates the number of transistors on microprocessors to reach $1.1$ million is $1989$.