In $y=-3x+k$, think separately: multiplying by $-3$ affects the range, but adding $k$ does not.

Once you know the median of A, apply $y=-3x+k$ to that median to express the median of B in terms of $k$.

You are told median(B) equals range(B). Set your two expressions equal and solve for $k$.

Enter the list of values from the dot plot:

A={1,2,2,3,3,3,4,4,5,5}

Type k=0 to create a slider.

Then define the transformed list:

B=-3A+k

Compute median(B) and max(B)-min(B).

Adjust the slider $k$ until median(B) equals max(B)-min(B), then read off the corresponding value of $k$.

There are 10 values, so the median is the average of the 5th and 6th values in the ordered list.

From the dot plot, the ordered values are

$1, 2, 2, 3, 3, 3, 4, 4, 5, 5$.

The 5th and 6th values are both $3$, so the median of A is $3$.

The minimum value is $1$ and the maximum value is $5$, so the range of A is

$5-1=4$.

Adding $k$ shifts every value by the same amount, so it does not change the range.

Multiplying every value by $-3$ scales all distances by $3$, so it multiplies the range by $3$.

Therefore, the range of B is $3\cdot 4=12$.

For a linear transformation $y=ax+b$, the median transforms the same way as the data values. Here $y=-3x+k$, so

$\text{median}(B)=-3\cdot \text{median}(A)+k=-3(3)+k=-9+k$.

The problem states that the median of B equals the range of B, so

$-9+k=12$.

Adding $9$ to both sides gives $k=21$.

Therefore, the correct choice is $21$.