This question is from an old midterm I'm using to review for my exam coming up...I don't see where the error is made in calculations...Can someone please explain?

Bunter was asked the following sequence of questions. His answers appear in red. Determine

where, if at all, Bunter has made an error or if his justifications are incomplete.

Wooster derives utility from consuming two goods. Let xi denote the quantity if good I consumed. His preferences over bundles (x₁, X₂) are represented by the following utility function: U(x₁,x₂) = 3 ln x₁+ ln x₂ where ln stands for the natural logarithm.

The price per unit of good 1 is $5 and the price per unit of good 2 is $2 and Wooster's income is $100.

a) The government implements an income tax. Let t be the fraction of income that the Government extracts. Write down Wooster's budget constraint. Assume t= 0.20. Solve for the optimal demand of both goods and the amount collected in income tax.

Justification: Wooster's utility maximization problem:

max 3 ln x₁ + ln x₂

x_1, x₂

s.t. 5x₁ + 2x₂= 100 (10.2)

x₁,x₂ 0

The Lagrangian is:

L= 3 ln x₁ + ln x_{2 +} (805x₁2x₂)

and the FOCs are:

3/x₁ 5= 0

1/x₂ 2= 0

Substituting the second equation into the first gives us 5x₁= 6x₂.

Plugging this equation into the budget constraint, we have 5(6/5 x₂) + 2x₂ = 80. Solving, we have that x ₁ = 12, x₂= 10, and the amount collected in taxes is 100 (10.2) = 20.