My question is with a multi-part homework assignment that we got in class. The information given is :

Question

A factory manufactures widgets. The rate of production of widgets after t weeks is widgets/ week. The equation for this is dx/dt=30(1-(20/((t+20)^2))).

The actual questions are below.

1)Determine the rate at which widgets are produced at the start of week 1. Set up the expression but do not solve.

For this one I just plugged "1" into the given equation. So the number of widgets produced at the start of week 1 is equal to 30(1-(20/(((1)+20)^2))). I'm pretty sure I am correct in this.

2)Determine the number of widgets produced from the beginning of production to the beginning of the fifth week.

For this I set up a definite integral of the given equation with b=5 and a=0. I am unsure whether I should put the given equation into the definite integral or if because the given equation is a rate, if I need to first take the anti-derivative of the given equation and then put that into the definite integral from 0 to 5. This question is where I am unsure to proceed.

3)Determine the number of widgets produced from the beginning of the fifth week to the end of the ninth week.

I thought to set up a definite integral from 5 to 10 (10,  because it wants to the end of the ninth week). As I said above I am also not sure if I need to first take the anti-derivative of the given equation and then put that into the definite integral. I am also unsure about where to proceed with this question.

bobpursley · Answer

For this one I just plugged "1" into the given equation. So the number of widgets produced at the start of week 1 is equal to 30(1-(20/(((1)+20)^2))). I'm pretty sure I am correct in this>>Nope. for the start of week 1, use t=0. Reread what the dn/dt represents (END OF WEEK T) <> b should be 4, same reason as above. On 3, should be 4 to 9. So take the integral of the function. Then apply the limits correctly