Variant 4. The PPF of the country can be described by the equation y=sqrt (128-x^2)(xgeqslant 0 ygeqslant 0) . The country enters the world trade where prices are p_(x)=2,p_(y)=2 1. Derive the TPF equation. 2. Calculate the opportunity cost of producing x in the point x=5

1. To derive the TPF (Terms of Trade) equation, we need to find the ratio of the price of good x to the price of good y. Given that the prices are $p_{x}=2$ and $p_{y}=2$, the TPF equation is simply the ratio of these prices, which is $\frac{p_{x}}{p_{y}} = \frac{2}{2} = 1$.<br /><br />2. The opportunity cost of producing x in the point $x=5$ can be calculated by finding the amount of good y that must be given up to produce an additional unit of good x. This can be done by taking the derivative of the PPF equation with respect to x and evaluating it at $x=5$.<br /><br />The derivative of the PPF equation $y=\sqrt{128-x^{2}}$ with respect to x is:<br /><br />$\frac{dy}{dx} = \frac{-x}{\sqrt{128-x^{}$<br /><br /> at $x=5$ gives:<br /><br />$\frac{dy}{dx}\bigg|_{x=5} = \frac{-5}{\sqrt{128-5^{2}}} = \frac{-5}{\sqrt{128-25}} = \frac{-5}{\sqrt{103}}$<br /><br />Therefore, the opportunity cost of producing x in the point $x=5$ is $\frac{-5sqrt{$.