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

Answer:

-1.1

Explanation:

Data provided in the question:

Average temperature is rising at a rate, $\frac{dT}{dt}$ = 0.15°C/year

Rate of change  rainfall, $\frac{dR}{dt}$ = - 0.1 cm/year

$\frac{\delta W}{\delta T}$ = -2

$\frac{\delta W}{\delta R}$ = 8

Now,

we need to calculate $\frac{dW}{dt}$

since,

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

thus, Using the chain rule , we have

$\frac{dW}{dt}$ = $\frac{\delta W}{dT}\times\frac{dT}{dt}$ +  $\frac{\delta W}{dR}\times\frac{dR}{dt}$

on substituting the respective values, we get

$\frac{dW}{dt}$ = -2 × 0.15 +  8 × (-0.1)

or

$\frac{dW}{dt}$ = -0.3 - 0.8

or

$\frac{dW}{dt}$ = -1.1