MAT 137 — LEC 0601 Instructor: Alessandro Malusà TA: Julia Kim November 26th, 2020 Warm-up : Get yourself some paper and a camera, and be ready to make some drawings and share your work with others!
The classic farmer problem A farmer has 300 m of fencing and wants to fence off a rectangular field and add an extra fence that divides the rectangular area in two equal parts down the middle. What is the largest area that the field can have?
Fire You hear a scream. You turn around: Ale is on fire!! Save his life, and he’ll most certainly reward you... You’re in luck: you’re standing on the bank of a straight river, and you’re holding an empty bucket (why are you holding an empty bucket?). Ale is standing still (as every normal person does when they catch fire) 10 meters away from the bank, and you’re 5 meters away from the point P on the river which is closest to him. There is only one thing to do: you must run along the river until you’re close enough to where Ale is, fill up the bucket, and then run straight towards him with the water; however, once the bucket is full you’ll run at half your normal speed. How far from P should you stop to pick up the water in order to get to Ale as fast as possible?
Distance Find the point on the parabola y 2 = 2 x that is closest to the point (1 , 4).
Airplane The cost of fuel per hour for a certain airplane is proportional to the square of its speed and is $1200 per hour for a speed of 600 km / h . After every 5,000 hours flown, the aircraft must undergo an $8 million dollar safety inspection. What speed should the airplane fly at in order to achieve the lowest cost per kilometre?
Dominion Dominion is a board game where, among other things, players buy cards worth victory points. The player with the most victory points wins. It is your last turn and you can only buy “duchies” and “dukes”. A duchy is worth 3 victory points. A duke is worth as many victory points as duchies you have. Each duchy costs 3 coins, and each duke costs 3 coins. You have not bough any duke or duchy yet. If you have N coins, how many dukes and how many duchies should you buy?
A matter of perspective A painting in an art gallery has height h and is hung so that its lower edge is a distance a above your eye. How far from the wall should you stand to get the best view? Note : I am purposely not specifying what "best view" means. Try and work that out on your own!
Before next class... • Watch videos 6.6, 6.7, and 6.8, and then 6.9 and 6.5. • Download the next class’s slides (no need to look at them!)
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