Chapter 5.6, Problem 47E

a.

To determine

To find: the rate of growth of the total production y .

Answer to Problem 47E

The rate of growth is $0.09y$ .

Explanation of Solution

Given information:

  $y=uv$

  $\frac{du}{dt}=0.04u$

  $\frac{dv}{dt}=0.05v$

Calculation :

We have to calculate the rate of growth of the total production y ..

Therefore,

  $y=uv$

Differentiate with respect to t using product rule,

  $\frac{dy}{dt}=u\frac{dv}{dt}+v\frac{du}{dt}$

Putting the value of $\frac{dv}{dt}=0.05v$ and $\frac{du}{dt}=0.04u$ in above equation,

  $\begin{array}{l}\frac{dy}{dt}=u\left(0.05v\right)+v\left(0.04u\right)\\ \frac{dy}{dt}=0.05uv+0.04uv\\ \frac{dy}{dt}=0.09uv\\ \frac{dy}{dt}=0.09y\left[\text{Since, }y=uv\right]\end{array}$

Hence, the value of $\frac{dy}{dt}$ is $0.09y$ .

b.

To determine

To find: whether the total production is increasing or decreasing, and at what rate.

Answer to Problem 47E

The value of $\frac{dy}{dt}$ is $0.01y$ is increasing.

Explanation of Solution

Given information:

  $y=uv$

  $\frac{du}{dt}=-0.02u\left[\begin{array}{l}\text{Since, labour force is decreasing }\\ \text{at the rate of 2\%/year}\text{.}\end{array}\right]$

  $\frac{dv}{dt}=0.03v\left[\begin{array}{l}\text{Since, production per person is}\\ \text{increasing at the rate of 3\%/year}\text{.}\end{array}\right]$

Calculation :

We have to calculate whether the rate of the total production is increasing or decreasing

Therefore,

  $y=uv$

Differentiate with respect to t using product rule,

  $\frac{dy}{dt}=u\frac{dv}{dt}+v\frac{du}{dt}$

Putting the value of $\frac{dv}{dt}=0.03v$ and $\frac{du}{dt}=-0.02u$ in above equation,

  $\begin{array}{l}\frac{dy}{dt}=u\left(0.03v\right)+v\left(-0.02u\right)\\ \frac{dy}{dt}=0.03uv-0.02uv\\ \frac{dy}{dt}=0.01uv\\ \frac{dy}{dt}=0.01y\left[\text{Since, }y=uv\right]\end{array}$

Hence, the value of $\frac{dy}{dt}$ is $0.01y$ is increasing.