Answer Solution to Question 3 a Finding the Maximum Number of Beneficiaries We are given the function N t 3 t 2 45 t 16 0 t 10 where N t represents the number of people in thousands benefiting from the project t years after completion To find the maximum number of beneficiaries we need to find the critical points by differentiating N t and setting the derivative equal to zero Step 1 Compute the First Derivative N t d d t 3 t 2 45 t 16 N t 6 t 45 Step 2 Solve N t 0 6 t 45 0 t 45 6 075 Step 3 Verify Maximum Using Second Derivative N t d d t 6 t 45 6 Since N t 6 is negative this confirms that t 075 is a local maximum Step 4 Check Endpoints Evaluating N t at the endpoints N 0 3 0 2 45 0 16 16 N 10 3 10 2 45 10 16 300 45 16 239 Since N 0 16 is greater than N 10 and the local maximum at t 075 is within the range the maximum number of beneficiaries occurs at t 075 years b Linear Programming Model for Clothing Production Step 1 Define Variables Let x number of trousers produced y number of skirts produced z number of blouses produced Step 2 Formulate Constraints Given time constraints in minutes Cutting time 20 x 15 y 10 z 115 60 since 115 hours 115 60 minutes Sewing time 60 x 30 y 24 z 280 60 Packaging time 5 x 12 y 6 z 65 60 Converting to total available minutes Cutting 20 x 15 y 10 z 6900 Sewing 60 x 30 y 24 z 16800 Packaging 5 x 12 y 6 z 3900 Step 3 Solve for Integer Solutions We solve the system using linear programming methods I will now compute the values The optimal number of clothing items to be produced to fully utilize available labor hours has been computed and displayed Let me know if you need further details