Question

There are currently 441 dairy cows at Dancing Dairy Farm. Due to some considerations, the number of dairy cows is decreasing at the rate of 13 cows per year. Currently, each cow produces an average of 1157 gallons of milk per year, and production of milk is increasing at the rate of 39 gallons per cow per year. Use the product rule to determine the rate at which milk production at Dancing Dairy Farm is currently changing.

Answer 1

Answer:

the production of milk at the Dancing Dairy Farm is increasing by 2158 gallons/year

Step-by-step explanation:

Given the data in the question;

Let us represent the number of cows at the farm with x and

each cow produces y gallons of milk

Total mil production will be T.

so

T = x × y

now, differentiating with respect to t

dT/dt = d( xy )/dt

dT/dt = xdy/dt + ydx/dt

given that;

x = 441

y = 1157

dx/dt = -13

dy/dt = 39

so we substitute

dT/dt = ( 441 )( 39 ) + ( 1157 )( -13 )

dT/dt = 17199 - 15041

dt/dT = 2158

Therefore, the production of milk at the Dancing Dairy Farm is increasing by 2158 gallons/year