SAT Question 9833d0e3: Nonlinear Functions answer and explanation with Desmos solution

Choice C is correct. It's given that equation $y=a{b}^{\frac{x}{100}}$ represents the model, where $y$ is the estimated volume, in cubic centimeters, of a certain tree in a forest and $x$ is the tree's height, in centimeters. For an exponential equation of this form to represent a model where the value of $y$ increases as the value of $x$ increases, it must follow that the values of $a$ and $b$ are positive. It's also given that the tree's estimated volume increases by $13.9\%$ for every increase of $100$ centimeters in the tree's height. Therefore, for every $100$ -centimeter increase in the tree's height, the estimated volume of the tree is $1.139$ times larger. In other words, $a{b}^{\frac{x+100}{100}}=1.139a{b}^{\frac{x}{100}}$, which is equivalent to $a{b}^{\frac{x}{100}}{b}^{\frac{100}{100}}=1.139a{b}^{\frac{x}{100}}$. Since $a$ and $b$ are positive, dividing both sides of this equation by $a{b}^{\frac{x}{100}}$ yields $b^{\frac{100}{100}}=1.139$, or $b=1.139$. Therefore, the value of $b$ is $1.139$.