[ROLE PLAYING: OPTIMIZATION] ( A WORKSHEET ON SECTION 4.6 ) (Due Friday 3/11)

Transcription

1 [ROLE PLAYING: OPTIMIZATION] ( A WORKSHEET ON SECTION 4.6 ) (Due Friday 3/11)

2 SCENARIO I Role: Adventurer/Rancher/Hero of time Problem: You want to enclose a rectangular plot of land so that your horse, Epona, can graze and run around in. You ve been budgeting for weeks, and can finally afford 2400 ft of fencing, so naturally you want to know what the largest area of land you can enclose is. To maximize your area, you also plan on using a straight river as one side of the field. What dimensions would produce the largest area for Epona?

3 SCENARIO II Role: Beverage company businessman/businesswoman Problem: You ve been working on overthrowing Coca-cola and Pepsi for years, and have finally come up with the idea of the century: Phoenix Down. It s a lemon-lime (square)soft drink dyed golden akin to the color of champagne. Your slogan is: You only die for real when you re out of Pheonix Down it s a work in progress. In any case, you want your cans to be cylinders, made from a thicker steel than cheap aluminum. Find the dimensions that will minimize the cost of the steel to manufacture the cans if you want your cans to hold 10oz.

4 SCENARIO III Role: Architect Problem: A client asks you to put a Norman window 1 in his house. If the Perimeter of the window is 30ft, what is the maximum amount of light the window could allow? 1 A Norman window has the shape of a rectangle surmounted by a semicircle (the diameter of the semicircle is coincident with the top width of the rectangular part of the window).

5 *SCENARIO IV Role: Number game fanatic Problem: You love number games! Someone has asked you two number game questions, and you want to solve both of them. 1. Find two positive numbers whose sum is 9 so that the product of one of the numbers with the square of the other is a maximum. 2. The sum of two numbers is 16. What is the smallest possible value of the sum of their squares?

6 *SCENARIO V Role: Adventurer/Rancher/Hero of time Problem: Now that Epona is taken care of in her new grazing pen, it s time to build an area for your Kakariko Cucco. The Kakariko Cucco requires at least 250 ft 2 of space it can be very dangerous if it feels too confined. If the area you want to fence is another rectangle, what would be the least amount of fencing required to construct a 250 square foot space?

7 *SCENARIO VI Role: Bear Grylls Problem: Your lost in the wilderness, and the sun is setting. You want to show your viewers how to survive in the wild, and today s challenge is crossing a river, to get to a location you feel would be safest to set up camp for the night. The river is 1.3 kilometers wide, and your campsite location is 8 kilometers downstream on the opposite bank. You build a makeshift raft out of branches and some rope, and reckon that you could row at a speed of 6km/h, but you also know that you can run at around 8km/h. You want to know what the fastest way to your campsite location is. One option is rowing directly across the river (say, to point A) and then running downstream to your campsite (we ll call it B). Another option is that you could simply row directly to B. A third option is that you could row to some point between A and B, then run the remaining downstream distance. How can you minimize travel time? Hint: remember that time = distance rate.

8 *SCENARIO VII Role: Carpenter Problem: After finishing a project, you have 1200 cm 2 of wood left over. Suppose you want to make a box to hold random nails and screws inside of such that the base of this box is a square, and there is no top. What is the largest possible volume you could construct?

9 *SCENARIO VIII Role: Math student Problem: Find the point (x, y) on the graph of y = x that is nearest the point (4, 0). Hint: recall that the distance formula between two points is: d = (x 2 x 1 ) 2 + (y 2 y 2 ) 2

10 *SCENARIO IX Role: Math student Problem: Find the dimensions of a rectangle of largest area which can be inscribed in the closed region bounded by the x-axis, y-axis, and the graph of y = 2 ( 1 2 ) x2.

11 *SCENARIO X Role: Landlord Problem: You manage a 100-unit apartment complex and know from experience that all 100 units will be occupied if you set the rent at $800 a month. A market survey you recently read suggests that, on average, one additional unit will remain vacant for each $10 increase in the monthly rent. What rent should you charge to maximize revenue? Hint: the function for revenue is R(x) = (amount per unit)(number of units). Find linear equations for each of those pieces in terms of x.

12 *SCENARIO XI Role: Snow cone vendor Problem: You sell the best snow cones on the block and you re proud of it. You serve them in coneshaped paper drinking cups, but you ve recently run out of them! If the cups are to be made to hold 27 cubic centimeters of ice, find the height and radius of the cup that will use the least amount of paper.