Since \(P_x = 8\)\(P_y = 4\)    , the slope of the budget curve is equal to -2. The optimum in consumption is reached when the marginal rate of substitution is equal to the slope of the budget curve. We know that:
\(MRS_{xy} = \frac{\Delta Y}{\Delta X} = - \frac{90}{X^2}\)
And we want to solve the equation:
\(-2 = - \frac{90}{X^2}\)
\(X^2 = 45\)
\(X = 45^{\frac{1}{2}}\)
We substitute this result in the utility function \(U(X ; Y) = XY = 90\):
\(\sqrt{45}Y = 90\)
\(Y = \frac{90}{\sqrt{45}}\)
\(Y = 2\sqrt{45}\)
Then we substitute the results for Y and X in the budget constraint:
\(I = 8X + 4Y\)
\(I = 8\sqrt{45} + 4*2\sqrt{45}\)
\(I = 16\sqrt{45}\)
So the income necessary so that the budget constraint is tangent to the indifference curve for which \(U(X ; Y) = 90\) is equal to \(16\sqrt{45}\), i.e. for the consumer to attain 90 utils of satisfaction at our given prices.
[PS]: To solve this question, we could use a Lagrangian so as to "maximize income." We find the same results with both methods.
2)
We have \(U(X_1 ; X_2) = X_1X_2 ; I=60 ; P_{X_1} = 5 ; P_{X_1} = 3\)
The budget constraint is equal to: \(60 = RX_1 + 3X_2\)
Let's set the Lagrangian:
\(\mathcal{L} = X_1X_2 - \lambda[60 - (5X_1 + 3X_2)]\)
\(\frac{dL}{dX_1} = 0 = X_2 - \lambda 5\)
\(\lambda = \frac{X_2}{5}\)
\(\frac{dL}{dX_2} = 0 = X_2 - \lambda 3\)
\(\lambda = \frac{X_1}{3}\)
\(\frac{dL}{d\lambda} = 0 = 60 - (5X_1 + 3X_2)\)
\(60 = 5X_1 + 3X_2\)
\(\frac{X_2}{5} = \lambda = \frac{X_1}{3}\)
\(3X_2 = 5X_1\)
\(X_2 = \frac{5}{3}X_1\)
We substitute this result in \(\frac{dL}{d\lambda} \) .
\(60 = 5X_1 + 3\frac{5}{3}X_1\)
\(X_1 = 6\)
\(X_2 = 10\)
3)
We have \(U(X ; Y) = 2XY ; I = 40 ; P_y = 4 ; P_x = 8\)
a)
The demand curve of  X correspond to the quantity of X demanded with respect to the price of X. We therefore consider \(P_x\) as a variable.
Let's set the Lagrangian:
\(\mathcal{L} = 2XY- \lambda[40 - 4Y - P_x X]\)
\(\frac{dL}{dX} = 0 = 2Y - \lambda P_x\)
\(\lambda = \frac{2Y}{P_x}\)
\(\frac{dL}{dY} = 0 = 2X - \lambda 4\)
\(\lambda = \frac{2}{4}X\)
\(\frac{dL}{d\lambda} = 0 = 40 - 4Y - P_xX\)
\(40 = 4Y + P_xX\)
\(\frac{2Y}{P_x} = \lambda = \frac{2}{4}X\)
\(2P_xX = 8Y\)
\(Y = \frac{1}{4}P_xX\)
Substituting in \(\frac{dL}{d\lambda} \), we have:
\(40 = 2P_xX\)
\(X = \frac{20}{P_x}\) is the demand curve for X.
b)
A 25% tax on the initial price of X means that the after tax price is of 10$ since we know that the pre-tax price is of 8$.
We know the demand curve for X from question (a):
\(X = \frac{20}{P_x}\)
So before the tax, since \(P_x = 8\), the quantity X demanded is equal to:
\(X = \frac{20}{8} = 2.5\)
The quantity demanded after the 25% tax is equal to:
\(X = \frac{20}{10} = 2\)
The budget constraint is equal to:
\(40 = 4Y + P_xX\)
So before the tax, \(P_x = 8\) and therefore:
\(Y = \frac{40 - 2.5*8}{4} = 5\)
After the tax, we have \(P_x = 10\):
\(Y = \frac{40 - 2*10}{4} =5\)
Since the utility function is \(U(X ; Y) = 2XY\), the consumer's utility is equal to 25 before the tax and 20 after the tax.
c)
After the tax, \(X = 2\), and since the tax yields 25% of the initial price of 8$. Therefore, the tax yielded 4$ of revenue. SInce \(I = 40\), the income tax has to be of 10% so as to yield the same amount of revenue. After tax income will be of 36$.
From question (a), we know that: 
\(X = \frac{0.5*I}{P_x}\)
Hence:
\(X = \frac{0.5*36}{8} = 2.25\)
We also know that:
\(Y = \frac{1}{4}P_xX\)
\(Y = \frac{1}{4}*8*2.25 = 4.5\)
We can theoretically account for this differences because in (b), you have a change in the relative price of X compared to Y and a readuction of real income as the after tax price of X increases. In the case of the income tax in question however, you only have an income effect. the relative price of X and Y is not changing.
d)
We know that the utility function is \(U(X ; Y) = 2XY\)
With the sale tax, the individual consumes 2 of X and 5 of Y. Therefore, \(U(X ; Y) = 20\)
With the income tax, the individual consumes 2.25 of X and 4.5 of Y. Therefore \(U(X ; Y) = 20.25\)
This means that the two goods in question are complementary enough so that an Income tax will be preferred to a tax on X.
e)
If we have 50 consumers identical to Mike and with the same budget constraint, then the demand of X is multiplied by 50. When the price of X is equal to 8$, the quantity of X demanded is of 2.5. That means that the total market demand for X is equal to 75.
f)
Lambda is the Lagrange multiplier and approximates the marginal impact on the objective function caused by a change in the constant of the constraint. In our exemple, \(\lambda\) approximates the impact of a change in income on total utility, or, to put it differently, approximates the marginal utility of money.
4)
We have \(X = 0.99I - 2.5P_x - 2.51P_y\)
A normal good is a good for which the quantity demanded increases when income increases. Hence since \(\frac{dX}{dI} = 0.99\), then X is a normal good.
\(\frac{dX}{dP_x} = -2.5\) which means that the demand curve is negatively sloped -i.e. an increase in \(P_x\) reduces the amount of X demanded.
If X and Y are substitute, then we would expect an increase in \(P_y\) to increase the demand for X. Since \(\frac{dX}{dP_x} = -2.51\), this means that X and Y are complementary, -i.e. the increase in the price of one good leads to a decrease in the demand of the other good.