My sister, the teacher, celebrates Shahrukh and Amitabh in her own special way. On their birthdays every year, her homework problems have a very special theme. Also, there is background music playing during class. And depending on the ethnic make-up of the group, they either really get into it (like when she was teaching in New Jersey and the class was 2/3rds desi) or are super confused. Anyway, this year she is letting me share with you all, so we can all learn about Shahrukh, and also math.
(1) Rahul Khanna and Anjali Sharma are best friends. Whenever they fight, they make up by tweaking each others’ noses. See:
(a) Anjali’s nose is shaped like 100 x2 y2 > z > 10 x2/10 y2/10 for 10 <
x < 10 and 0 < y < 10. Use Mathematica to draw a picture of Anjali’s nose.
(b) Use a triple integral to find the volume of Anjali’s nose.
(c) Rahul’s nose has a more dramatic shape. Use Mathematica to come up with a picture of Rahul’s nose.
(d) Use a triple integral to find the volume of Rahul’s nose.
(2) Rahul Raichand is hurrying home to his family’s estate for Diwali. Luckily, he has his own helicopter. The family estate is located at x = 0, y = 0, z = 0 and his helicopter is
at position x = 1, y = 1, z = 10. See:
(a) What is Rahul’s initial position in cylindrical coordinates?
(b) What is the family estate’s position in cylindrical coordinates?
(c) For safety reasons, air traffic control wants to create a no-fly-zone while Rahul travels. What would be a good region (in cylindrical coordinates) to block off from other flights?
(d) What is the total volume of that region? Your answer should use a triple integral.
(3) Raj and Meera have fallen in love. Swept away by emotion, they sing to each other in Iceland. The ice sheet they’re standing on has strength related to its distance from the North Pole (which is located at x = 0 y = 0). At distance x2 + y2 away, its strength is 100/ x2 + y2 pounds / square inch. See:
(a) Rewrite the strength function in polar coordinates
(b) Safety first! They decide to build a candy-corn-shaped platform to support their weight. If their total weight is 300 pounds, what platform would be big enough to hold them? Where is it located? Note: There are multiple correct answers.
(4) Shah Rukh Khan has been through several hairstyles over the years. Find the volume of each of them.
(a) Main Hoon Na His scalp is a circle of radius 1 centered at (0, 0) and at position (x, y) his hair has length (2 x4 y4). See:
(b) In Koyla His mullet forms a half-cylinder and the hair on top of his head is shaped like 10 − x2 − y2.
(5) You have decided to celebrate Shah Rukh Khan’s birthday, as is only right, with birthday cake. You’re prone to dropping things. Fortunately, you remember the formula for center-of-mass. Where is the best place to hold
(a) a 9x9x2 marble cake whose density changes according to the chocolate and vanilla swirls: the density function is sin πx + cos πy + xy?
(b) A vanilla cake made by taking a sphere of radius one and cutting it in half in all three directions, to get an octant?
(c) Are you hungry for cake now? What kind would you make to celebrate SRK?
And, since she brought up cake, here is the one I just bought and brought back to the office (very dangerous having an office around the corner from a bakery. And having it be only slightly more to buy a whole cake versus a slice)