Winter 2023 | Calculus and Vectors | Grade 12 Performance Task #5 - Optimization and R.Rs Instructions: I Pick any 7 and provide a full solution. Round to 4 decimal places whenever necessary. I For questions with \"', use the name cypher to replace nwith the rst letter in your name. I Each question is graded out of 5 marks. 1. Find a positive number such that the sum of the number and its reciprocal is as small as possible. 2. Find the dimensions of a rectangle with area 1000 m2 whose perimeter is as small as possible. 3. A model used for the yield Y of an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units) is: Y = kN r (1 + N2) where k is a positive constant. What nitrogen level gives the best yield? 4. '\" A box with a square base and open top must have a volume of (32000 + TL) cm3. Find the dimensions of the box that minimize the amount of material used. 5. \"' If (1200 + 11) cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box. 6. \"' A cone-shaped paper drinking cup is to be made to hold (27 + TL) cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper. T. A funnel in the shape of an inverted cone is 30 cm deep and has a diameter across the top of 20 cm. Liquid is owing out of the funnel at the rate of 12 cm3lsec. At what rate is the height of the liquid decreasing at the instant when the liquid in the funnel is 20cm deep? 8. Gas is escaping a spherical balloon at the rate of (4 + 71/100) cm3 per minute. How fast is the surface area shrinking when the radius is (10 + TI.) cm? 9.A boy is ying a kite on a string. The kite is 120 ft above the ground and the wind is blowing the kite horizontally away from him at 6 ftr'sec. At what rate must he let out the string when 130 ft of string has been let out? 10. A right circular cylinder of constant volume is being attened. At the moment when its radius is 3 cm, the height is 4cm and the height is decreased at the rate of (0.1 + 71/100) cmlsec. At that moment, what is the rate of change of the radius? 11. A 178 cm tall man is walking towards a lamppost which is 310 cm high. The lamp casts a shadow behind him. He notices that his shadow gets shorter as he moves closer to the lamp. He is walking at 2.2 mfsec. When he is 2 meters from the lamppost, how fast is the length of his shadow decreasing? 12. '\" Water is draining from a cylindrical tank at (2 + 71/100) Lisec. Assume that the water tank has a diameter of 42 cm. How fast is the surface dropping