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To produce batteries, a manufacturer uses x units of chemical A and y units of chemical B. The amount P of pollution washed i

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Answer #1

\text{ Given pollution function is }

P(x, y) =0.05x^2+0.009xy+0.07y^2

\text{ To find the rate at which the amount of pollution washed into the cleaning water is changing }

\text{as the number of units of chemical B changes. }

\text{ We will Calculate }P_y(x,y )

P_y(x, y) =\frac{\partial }{\partial y}\left (0.05x^2+0.009xy+0.07y^2 \right )

=> P_y(x, y) =\frac{\partial }{\partial y}\left (0.05x^2 \right ) +\frac{\partial }{\partial y}\left (0.009xy \right )+\frac{\partial }{\partial y}\left (0.07y^2 \right )

=> P_y(x, y) =0+0.009x\:\: \frac{\partial }{\partial y}\left (y \right )+0.07\:\:\frac{\partial }{\partial y}\left (y^2 \right )

=> P_y(x, y) =0.009x(1)+0.07(2y)

=> P_y(x, y) =0.009x+0.14y

\text{ Calculating }P_y(10,19 )

P_y(10, 19) =0.009(10)+0.14(19) =2.75

=> P_y(10, 19)=2.75>0

=> \text{ This rate is changing at an increasing rate}

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