4. Katie likes to paint and sit in the sun. Her utility function is U(P, S) = 3PS + 6P, where P is the number of paint brushes and S is the number of straw hats. The price of a paint brush is $1 and the price of a straw hat is $5. Katie has $50 to spend on paint brushes and straw hats.

a. Solve Katie’s utility-maximization problem using a Lagrangian.

b. How much does Katie’s utility increase if she receives an extra dollar to spend on paint brushes and straw hats?