please, if you can explain on a piece of paper, I would appreciate it. I have tried doing most questions, but some are just too tough

MCV4U OPTIMIZATION PROBLEMS art A: Working with Equations 1. The height, h, of a ball & seconds after being thrown into the air is given by the function h(t) = -4.912 + 19.6t + 2. Find the maximum height of the ball. (ANS: 21.6 m) 2. At G&W Industries, it has been shown that the number of gizmos an employee can produce each day can be represented by the equation N(t) = -0.05t2 + 3t + 5, where t in the number of years of experience the employee has, and 0 S t S 40. How many years of experience does it take to achieve maximum productivity? (ANS: 30) art B: Numbers 3. Find two numbers whose difference is 150 and whose product is a minimum. (ANS: 75 and -75) 4. Find two positive numbers with a product of 200, such that one number and twice the second number is as small as possible. (ANS: 20 and 10) 5. Find two numbers with a sum of 20 and whose product is a maximum. (ANS: 10 and 10) 6. Find a positive number such that the sum of the square of the number and its reciprocal is a minimum. (ANS: 0.794) art C: Area/Perimeter 7. A rectangle has a perimeter of 100 cm. What length and width should it have so that the area is a maximum. (ANS: 1 = 25 cm and w = 25 cm) Ans: 1= 25 cm and W = 25 cm 8. The lifeguard at a public beach has 400 m of rope available to lay out a rectangular restricted swimming area using the straight shoreline as one side of the rectangle. a) If she wants to maximize the swimming area, what will the dimensions of the rectangle be? (ANS: 100 m by 200 m) b) To ensure the safety of swimmers, she decides that nobody should be more than 50 m from shore. What should the dimensions of the swimming area be with this added restriction? (ANS: 50 m by 300 m) 9 . A farmer wants to fence an area of 750 000 m in a rectangular field and divide it in half with a fence parallel to one of the sides of the rectangle. How can this be done so as to minimize the cost of the fence? (ANS: 500v2 m by 750 v2m) 10. A Norman window has the shape of a rectangle capped by a semicircular region as shown. If the perimeter of the window is 8 m, find the width of the window that will admit the greatest amount of light. (ANS: "7+4 = 2, 24 m) 11. A 1 kilometre racetrack is to be built with two straight sides and semicircles at the ends (as in the figure), Find the dimensions of the track that encloses the maximum area. (ANS: circular with r = ,_ km) 12. A rectangular pen is to be built with 1200 m of fencing. The pen is to be divided into three parts using two parallel partitions. Find the maximum possible area of the pen. (ANS: 45000 m?) 13. Two pens with one common side are to be built with 60 m of fencing. One pen is to be square, the other rectangular, as shown. Find the dimensions that maximize the total area. (ANS: 10 m by 5 m)