To solve this problem, we first need to define the decision variables and the objective function. Let's denote the number of cups of carrots as \(x\) and the number of cups of spinach as \(y\). The objective is to minimize the total cost, which can be expressed as \(5x + 3y\), given that the cost per cup of carrots is $5 and the cost per cup of spinach is $3.

The constraints are based on the nutritional requirements:
1. For biotin: \(x + 2y \geq 20\) (since one cup of carrots contains 1 gram of biotin and one cup of spinach contains 2 grams of biotin, and he needs at least 20 grams).
2. For folate: \(3x + 1.5y \geq 20\) (since one cup of carrots contains 3 grams of folate and one cup of spinach contains 1.5 grams of folate, and he needs at least 20 grams).

Additionally, \(x \geq 0\) and \(y \geq 0\), because the number of cups cannot be negative.

Now, let's express this problem in Gurobi code using Python:

```python
from gurobipy import *

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

# Define variables
x = m.addVar(vtype=GRB.CONTINUOUS, name="carrots", lb=0)
y = m.addVar(vtype=GRB.CONTINUOUS, name="spinach", lb=0)

# Set the objective function
m.setObjective(5*x + 3*y, GRB.MINIMIZE)

# Add constraints
m.addConstr(x + 2*y >= 20, "biotin_requirement")
m.addConstr(3*x + 1.5*y >= 20, "folate_requirement")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Carrots: {x.x} cups")
    print(f"Spinach: {y.x} cups")
    print(f"Total Cost: ${5*x.x + 3*y.x:.2f}")
else:
    print("No optimal solution found.")
```