To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. Let's define the variables as follows:

- $x_1$ represents the number of youth doses.
- $x_2$ represents the number of adult doses.

The objective function is to maximize profit. Given that youth doses are sold for a profit of $5 and adult doses for a profit of $3, the objective function can be represented as:

\[ \text{Maximize:} \quad 5x_1 + 3x_2 \]

The constraints based on the problem description are:

1. The total amount of extract used does not exceed 5000 grams:
\[ 20x_1 + 35x_2 \leq 5000 \]
2. At least three times as many youth doses are needed than adult doses:
\[ x_1 \geq 3x_2 \]
3. A minimum of 10 adult doses need to be made:
\[ x_2 \geq 10 \]
4. Non-negativity constraints (since we cannot produce a negative number of doses):
\[ x_1 \geq 0, \quad x_2 \geq 0 \]

In symbolic notation, this problem can be represented as:

```json
{
    'sym_variables': [('x1', 'youth doses'), ('x2', 'adult doses')], 
    'objective_function': '5*x1 + 3*x2', 
    'constraints': [
        '20*x1 + 35*x2 <= 5000',
        'x1 >= 3*x2',
        'x2 >= 10',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

Now, let's write the Gurobi code to solve this problem:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="youth_doses")
x2 = m.addVar(vtype=GRB.INTEGER, name="adult_doses")

# Set the objective function
m.setObjective(5*x1 + 3*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(20*x1 + 35*x2 <= 5000, "extract_limit")
m.addConstr(x1 >= 3*x2, "youth_to_adult_ratio")
m.addConstr(x2 >= 10, "adult_doses_minimum")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution found: {x1.varName} = {x1.x}, {x2.varName} = {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found")
```