Question 519·Hard·Math

Each dot represents one day, so the frequencies are $1, 3, 4, 3, 1$ at $2, 4, 6, 8, 10$ deliveries. Enter those paired columns in Desmos. A mean is the total divided by the number of values, so type $\frac{\operatorname{total}(x_1y_1)}{\operatorname{total}(y_1)}$. Desmos shows $6$ deliveries per day. Multiplying each value by its frequency counts every dot.

The app recorded one extra delivery on every day, so each value $x_1$ becomes $x_1-1$. Type $\operatorname{total}((x_1-1)y_1)$ in Desmos. It shows $60$ deliveries for the corrected 12 days; the frequencies $y_1$ make sure the correction reaches every day.

The added day contributes $k$ to the corrected total of $60$ and raises the count from $12$ to $13$. So the new mean must satisfy $\frac{60+k}{13}>6$. The denominator is $13$, not $12$, because the new day belongs in the new data set.

To find the boundary, replace $>$ with equality. Type $\frac{60+k_1}{13}\sim6$ in Desmos: the subscript lets Desmos fit $k_1$ instead of making a slider, and $\sim$ asks it to make the sides equal. Under PARAMETERS, Desmos shows $k_1=18$. At that value, the new mean only matches the original mean.

Increasing $k$ increases the new mean, so greater than the original mean requires $k>18$. The equal-mean boundary itself does not count. The least integer above $18$ is $19$ deliveries on the added day. Choice B.