Answer to Question #320382 in Calculus for koi

Question

Suppose the output at a certain factory is 𝑄 = 2𝑥 3 + 𝑥 2𝑦 + 𝑦 2 units, where 𝑥 is the number of hours of skilled labour used and 𝑦 is the number of hours of unskilled labour. The current labour force consists of 30 hours of skilled labour and 20 hours of unskilled labour. Use calculus to estimate the change in unskilled labour y that should be made to offset a 1-hour increase in skilled labour x so that output will be maintained at its current level

Answer 1

$Q=2x^3+x^2y+y^2\\dQ=\left( 6x^2+2xy \right) dx+\left( x^2+2y \right) dy\\x=30,y=20,dx=1,dQ=0\Rightarrow \\\Rightarrow 0=6600\cdot 1+940dy\Rightarrow dy=-\frac{6600}{940}=-7.02128$