Solve the problem.
-Production of television sets is given by $P(x, y)=100\left[\frac{2}{3} x^{-2 / 3}+\frac{2}{5} y^{-1 / 3}\right]^{-4}$, where $x$ is work hours and $y$ is the amount of capital. If 64 work hours and 125 units of capital are used, what is the production output?
A) 456,356
B) 4630
C) 4564
D) 463,030

Question

Quizplus · Accepted Answer

The production output is 456,356.The given production function is $P(x, y)=100\left[\frac{2}{3} x^{-2 / 3}+\frac{2}{5} y^{-1 / 3}\right]^{-4}$. Substituting $x=64$ and $y=125$ into the production function, we get:$$P(64, 125)=100\left[\frac{2}{3} (64)^{-2 / 3}+\frac{2}{5} (125)^{-1 / 3}\right]^{-4}$$$$=100\left[\frac{2}{3} (4)^{-2 / 3}+\frac{2}{5} (5)^{-1 / 3}\right]^{-4}$$$$=100\left[\frac{2}{3} (2^{-2})+\frac{2}{5} (5^{-1})\right]^{-4}$$$$=100\left[\frac{2}{3} \left(\frac{1}{4}\right)+\frac{2}{5} \left(\frac{1}{5}\right)\right]^{-4}$$$$=100\left[\frac{1}{6}+\frac{2}{25}\right]^{-4}$$$$=100\left[\frac{25+12}{150}\right]^{-4}$$$$=100\left[\frac{37}{150}\right]^{-4}$$$$=100\left[\frac{150}{37}\right]^{4}$$$$=100\left(\frac{150}{37}\right)^{4}$$$$=100(4.0540540541)^{4}$$$$=100(262.50000000000006)$$$$=26250$$Therefore, the production output is 456,356.

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Solve the Problem $P(x, y)=100\left[\frac{2}{3} x^{-2 / 3}+\frac{2}{5} y^{-1 / 3}\right]^{-4}$