Question

To produce batteries, a manufacturer uses x units of chemical A and y units of chemical B. The amount P of pollution washed into the cleaning water is given by the pollution function P(x, y) = 0.05x2 +0.009xy +0.07y2 If the company is currently using 10 units of chemical A and 19 units of chemical B, find the rate at which the amount of pollution washed into the cleaning water is changing as the number of units of chemical B changes. Determine if this rate is changing at an increasing or decreasing rate. P (10, 19) Preview This rate is changing at a(n) Select an answer v rate.

Answer 1

$\text{ Given pollution function is }$

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

as the number of units of chemical B changes.

We will Calculate Py(x, y)

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

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

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

$\text{ Calculating }P_y(10,19 )$

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