To find the height of the package with maximum volume, we need to analyze the given conditions and use mathematical optimization techniques.

Let's break down the problem step by step:

1. Understanding the Constraints:
- The package should be in the form of a right circular cylinder.
- The sum of the height and the diameter of the base should not exceed 20 cm.

2. Formulating the Problem:
- We need to find the height of the cylinder that will result in the maximum volume.
- Let's assume the height of the cylinder as 'h' and the diameter of the base as 'd'.
- The volume of a cylinder is given by V = πr²h, where r is the radius of the base.

3. Expressing Constraints in Terms of Variables:
- The sum of the height and the diameter of the base should not exceed 20 cm.
- Since the diameter is twice the radius, we can express this constraint as h + 2r ≤ 20.

4. Relating Variables:
- We can relate the height 'h' and the radius 'r' using the formula for the diameter of the base: d = 2r.
- Rearranging this equation, we get r = d/2.

5. Expressing Volume in Terms of Single Variable:
- Substituting the value of r in the volume equation, we get V = π(d/2)²h.
- Simplifying further, V = (π/4)d²h.

6. Finding the Maximum Volume:
- To find the maximum volume, we need to maximize the expression (π/4)d²h.
- Since d is twice the radius, we can express it as d = 2r = 2(h/2) = h.
- Substituting this value, the expression becomes V = (π/4)h²h = (π/4)h³.
- Now, we need to find the value of h that maximizes this expression.

7. Applying Optimization Techniques:
- To find the maximum value of a function, we differentiate it with respect to the variable and equate it to zero.
- Differentiating V = (π/4)h³ with respect to h, we get dV/dh = (3π/4)h².
- Setting dV/dh = 0, we find h² = 0, which implies h = 0.

8. Analyzing the Result:
- Since the height cannot be zero, we need to consider the boundary conditions.
- The sum of the height and the diameter of the base should not exceed 20 cm.
- Let's consider the case when the sum is exactly 20 cm: h + d = 20.
- Substituting d = h, we get h + h = 20, which implies 2h = 20.
- Solving this equation, we find h = 10 cm.

Therefore, the height of the package with maximum volume that will be accepted by the shopkeeper is 10 cm, which corresponds to option B) 20/3 cm.