To solve question 1, you correctly plugged in 1 into the given equation to find the rate at which widgets are produced at the start of week 1. The expression you set up is 30(1-(20/((1+20)^2))). While you didn't simplify the expression further, you have correctly set it up.
For question 2, you want to determine the number of widgets produced from the beginning of production to the beginning of the fifth week. To do this, you need to integrate the given equation over the interval from 0 to 5. Since the given equation dx/dt represents the rate of production, you do not need to take any additional steps such as finding the antiderivative. You can directly integrate the given equation with respect to t as follows:
∫[0,5] 30(1-(20/((t+20)^2))) dt
To find the solution to this integral, you can use any integration technique or software tool you prefer.
For question 3, you want to determine the number of widgets produced from the beginning of the fifth week to the end of the ninth week. Similar to question 2, you can set up a definite integral to find this value. Since you're looking for the number of widgets produced during the time interval from 5 to 9 (end of the ninth week), you can set up the following definite integral:
∫[5,9] 30(1-(20/((t+20)^2))) dt
Again, you do not need to take the antiderivative of the given equation since it already represents the rate of production. Just evaluate the integral using your preferred integration method or software tool to find the answer.
Note: Remember to evaluate the definite integrals using the limits of integration, in this case, [0,5] and [5,9].