Alright, let's break down this problem into easy-to-understand steps.
A) Set up the Lagrangian for this problem:
Sydney has 6 hours from 6 pm to midnight, so she has 6*60 = 360 minutes. Each episode of her favorite show on Netflix takes 30 minutes, and each homework takes 1 hour (or 60 minutes). We'll denote the number of homework Sydney does as H and the number of Netflix episodes she watches as N.
Her utility function is U(H,N) = 4HN, and she wants to maximize this utility.
She has a time constraint: 60H + 30N <= 360 (because each homework takes 60 minutes and each episode takes 30 minutes, and she has 360 minutes in total).
Now, let's introduce a Lagrange multiplier, which we'll call λ (lambda). The Lagrangian L for this problem is:
L = 4HN + λ(360 - 60H - 30N).
B) Solve for optimal H and N:
To find the optimal H and N, we need to find the values that maximize L. We do this by taking the partial derivatives of L with respect to H, N, and λ, and set them equal to 0:
∂L/∂H = 4N - 60λ = 0
∂L/∂N = 4H - 30λ = 0
∂L/∂λ = 360 - 60H - 30N = 0
From the first equation: 4N = 60λ -> N = 15λ
From the second equation: 4H = 30λ -> H = 7.5λ
Plug these into the third equation:
360 - 60(7.5λ) - 30(15λ) = 0
360 - 450λ - 450λ = 0
360 = 900λ
λ = 0.4
Now plug λ back into N = 15λ and H = 7.5λ:
N = 15(0.4) = 6
H = 7.5(0.4) = 3
So, Sydney should do 3 homework and watch 6 episodes to maximize her utility.
C) If the utility function was now U(H,N) = 10HN, how would your answer to (B) change?
Now, let's update the utility function: U(H,N) = 10HN.
The new Lagrangian is:
L = 10HN + λ(360 - 60H - 30N).
Take the partial derivatives and set them equal to 0:
∂L/∂H = 10N - 60λ = 0
∂L/∂N = 10H - 30λ = 0
Now, from the first equation: 10N = 60λ -> N = 6λ
From the second equation: 10H = 30λ -> H = 3λ
Plug these into 360 - 60H - 30N = 0:
360 - 60(3λ) - 30(6λ) = 0
360 = 270λ
λ = 360/270 ≈ 1.33
Plug λ back into N = 6λ and H = 3λ:
N ≈ 6(1.33) ≈ 8
H ≈ 3(1.33) ≈ 4
With the new utility function, Sydney should do around 4 homework and watch around 8 episodes to maximize her utility
.
In simple terms, when the utility function changed, the weight placed on watching Netflix and doing homework in the utility function increased. This means that Sydney finds it more beneficial to allocate more time to both activities.