Question

Wheat production W in a given year depends on the average temperature T and the annual rainfall R. Scientists estimate that the average temperature is rising at a rate of 0.15°C/year and rainfall is decreasing at a rate of 0.1 cm/year. They also estimate that, at current production levels, δW/δT = -2 and δW/δR = 8. Estimate the current rate of change of wheat production, dW/dt.

Answer 1

-1.1

Step-by-step explanation:

Data provided in the question:

Average temperature is rising at a rate,
`(dT)/(dt)` = 0.15°C/year

Rate of change rainfall,
`(dR)/(dt)` = - 0.1 cm/year

`(\delta W)/(\delta T)` = -2

`(\delta W)/(\delta R)` = 8

Now,

we need to calculate
`(dW)/(dt)`

since,

The wheat production (W) is dependent on the rainfall (R) and the Temperature (T)

thus, Using the chain rule , we have

`(dW)/(dt)` =
`(\delta W)/(dT)*(dT)/(dt)` +
`(\delta W)/(dR)*(dR)/(dt)`

on substituting the respective values, we get

`(dW)/(dt)` = -2 × 0.15 + 8 × (-0.1)

or

`(dW)/(dt)` = -0.3 - 0.8

or

`(dW)/(dt)` = -1.1