Explanation:
To find out after how many of Joy's sessions she swims the same number of lengths as Ann, follow these steps:

1. Determine Ann's number of lengths per session:

Ann starts with 10 lengths and increases by 2 lengths per session until she reaches a maximum of 50 lengths. After reaching 50 lengths, she maintains this number.

- For Ann:
- In the 5th session, Ann swims:
\[
a_5 = 10 + (5 - 1) \times 2 = 10 + 8 = 18
\]
- In the 6th session, she swims:
\[
a_6 = 10 + (6 - 1) \times 2 = 10 + 10 = 20
\]
- And so on, until she reaches 50 lengths, which happens in the 21st session. From the 21st session onward, she swims 50 lengths.

2. Determine Joy's number of lengths per session:

Joy starts swimming 15 lengths and increases by 5 lengths per session.

- For Joy:
- In her first session (which is Ann's 5th session), Joy swims:
\[
j_1 = 15
\]
- In her second session, she swims:
\[
j_2 = 15 + 5 = 20
\]
- And so on, increasing by 5 each session.

3. Find the session where Joy's number of lengths equals Ann's number of lengths:

We need to match the lengths Joy swims to the number of lengths Ann swims.

- For Joy:
\[
j_n = 15 + (n - 1) \times 5
\]
- For Ann:
- Ann swims \(10 + (n - 1) \times 2\) lengths per session until the 21st session, where she swims a constant 50 lengths.

We set Joy's lengths equal to Ann's lengths:

- Before Ann swims 50 lengths:
\[
15 + (n - 1) \times 5 = 10 + (n - 5) \times 2
\]
Simplify:
\[
15 + 5n - 5 = 10 + 2n - 10
\]
\[
5n + 10 = 2n
\]
\[
3n = 10
\]
\[
n = 4
\]

After 4 sessions of Joy, her number of lengths matches Ann’s number of lengths.

Therefore, Joy will swim the same number of lengths as Ann after her 4th session.

The correct answer is D. 4.