Question

Alabama Instruments Company has set up a production line to manufacture a new calculator. The rate of production of these calculators after t weeks is

dx/dt=5000(1−100/(t+10)2)calculators/week

(Notice that production approaches 5000 per week as time goes on, but the initial production is lower because of the workers’ unfamiliarity with the new techniques.) Find the number of calculators produced from the beginning of the third week to the end of the fourth week.

Answer 1

The number of calculators is 4871

Explanation:

If we integrate dx/dt we get x, which is the number of calculators. To find the number of calculators between the beginning of third week to the end of fourth week (the beginning of fifth week), this integration must be evaluated at t between 3 and 5.

`x=\int\limits^5_3 {(dx)/(dt)} \, dt =\int\limits^5_3 {5000(1-(100)/((t+10)^2))} \, dt`

the result of the integration is:

`x=5000(t+(100)/(t+10))` to be evaluated between 3 and 5, which is:

`x=5000(5+(100)/(5+10))-5000(3+(100)/(3+10))=(190000)/(39)=4871.8`