SAT Question cf2be18d: Distributions, Center & Spread answer and explanation with Desmos solution

Choice C is correct. It’s given that the data set containing $a$, $26$, $29$, $b$, $31$, $47$, and $c$ is listed in ascending order and has a mean of $36$, a median of $29$, and a range of $72$. Since there are $7$ values in the given data set, it follows that the median is the value in the fourth position. It's given that $b$ is in the fourth position in the given data set, so it follows that $b=29$. Since the mean of the data set is $36$, it follows that $\frac{(a+26+29+29+31+47+c)}{7}=36$. Multiplying both sides of this equation by $7$ yields $a+26+29+29+31+47+c=252$, which is equivalent to $a+c+162=252$. Subtracting $162$ from both sides of this equation yields $a+c=90$. The range is the difference between the greatest and least values in a data set. Since $c$ is the greatest value and $a$ is the least value in the given data set, it follows that $c-a=72$, or $-a+c=72$. Adding the equations $a+c=90$ and $-a+c=72$ yields $2c=162$. Dividing both sides of this equation by $2$ yields $c=81$.