Pre-Calculus Applications Optimization Day 1

Transcription

1 Pre-Calculus Applications Optimization Day 1 Name Pd 1. The sum of 2 numbers is 8. A. write an equation for the sum. B. write a function for the product of the two numbers C. Graph the product function and find the numbers that maximize the product. 2. The sum of 2 numbers is 5. A. write an equation for the sum. B. write a function for the product of the two numbers C. Graph the product function and find the numbers that maximize the product. 3. The difference of 2 numbers is 2. A. write an equation for the difference. B. write a function for the product of the two numbers C. Graph the product function and find the numbers that minimize the product. 4. A rectangle has a perimeter of 20. What are the dimensions of the rectangle that maximize the area of the rectangle? 5. The sum of 2 numbers is 30, maximize the product 6. The difference of 2 numbers is 10, minimize the product. 7. Look at the values that maximize products in questions 1, 2, 4, and 6. Find a pattern. If the roots of a function you are maximizing are 0 and 12, where would the max occur? What does this mean geometrically?

2 Optimization Day 2 Name Pd 8. The sum of 2 numbers is 40, maximize the product 9. The perimeter of a rectangle is 40, maximize the area. 10. a rectangle has an area of 10, minimize the perimeter. 11. The difference of 2 numbers is 9, maximize the product. 12. A square prism has a surface area of 100, maximize the volume. 13. A square prism has a volume of 40, minimize the surface area? 14. A right triangle has a perimeter of 10. What are the 3 dimensions of the triangle that maximize the area? Hint: you must create 3 equations and narrow them down to 1 that has only x to determine area. 15. The sum of two numbers is n. Prove that maximum product will occur when both numbers are equal. Use you answer to question 7 from day 1 to help you. Hint:

3 Optimization Day 3 Name Pd 16. An arrow show vertically into the air from a crossbow reaches a maximum height of 484 feet after 5.5 seconds of flight. Let the quadratic function d(t) represent the distance above the ground (in feet) t seconds after the arrow is released. Part 1: find d(t) and state its domain. Part 2: Sketch the path of the arrow over time in detail. Part 3: At what times will the arrow be 250 feet above the ground? 17. An arrow show vertically into the air from a crossbow reaches a maximum height of 324 feet after 4.5 seconds of flight. Let the quadratic function d(t) represent the distance above the ground (in feet) t seconds after the arrow is released. Part 1: find d(t) and state its domain. Part 2: Sketch the path of the arrow over time in detail. Part 3: At what times will the arrow be 250 ft above the ground?

4 18. A square piece of cardboard, 8 inches on each side is used to make an open top box by cutting out a small square from each corner and bending up the sides. Part 1: Sketch the rectangle and the corner cutouts. Label everything with respect to x. Part 2: State an equation for the volume of the box with respect to x and state the domain. Part 3: Sketch the volume of the box over the defined domain in detail. Part 4: Analytically find the dimensions of the box of largest volume and the maximum volume. Sketch this box as the cardboard before it is folded AND after it is folded into its final shape. Before After

5 19. An open-top rectangular box is constructed from a 10- by 16- in. piece of cardboard by cutting squares of equal side length from the corners and folding up the sides. Part 1: Sketch the rectangle and the corner cutouts. Label everything with respect to x. Part 2: State an equation for the volume of the box with respect to x and state the domain. Part 3: Sketch the volume of the box over the defined domain in detail. Part 4: Analytically find the dimensions of the box of largest volume and the maximum volume. Sketch this box as the cardboard before it is folded AND after it is folded into its final shape. Before After

6 Optimization Day A horse breeder wants to construct a corral next to a horse barn 50 feet long, using the barn as one side of the corral. He as 250 feet of fencing available and wants to use all of it. Part 1: Express the area A(x) of the corral as a function of x and indicate its domain given that x 0. Part 2: Sketch the area in detail over the domain on the grid below. Part 3: Find the value of x that maximizes the area. Part 4: What are the dimensions of the corral that maximize the area? Part 5: Now you will repeat the scenario with two modifications: The amount of fence is 140 ft. The restraint on the domain is changed so that x -50. What does this restraint mean with respect to the side of the barn? Part 6: Sketch the area using the new domain. Part 7: Find the value of x that maximizes the area. Part 8: What are the dimensions of the corral that maximize the area? Part 9: What would you do if you were the building the fence?

7 21. You are on the leadership of your local workers union and have been offered a 2 year contract. You will receive a total of a 6% raise over the 2 years but you are allowed to describe how much of the 6% you receive each year. For example, you could choose 6% the first year and 0% the second or 4% the first year and 2% the second year. The raises must add to 6%. The current starting salary is $30,000. Part 1. Complete the table to understand the variations on the final money: Starting salary First year raise First year salary Second year raise Second year salary Total Money earned $30,000 0% $30,000 6% $31,800 $61,800 $30,000 1% $30,000 2% $30,000 3% $30,000 4% $30,000 5% $30,000 6% Part 2. What is the domain for this situation? Part 3.Write a function for the first year salary over time using x as the percent raise for the first year, graph the function for the domain, and maximize this graph. First Year Formula: Part 4.Write a function for the second year salary over time using x as the percent raise for the first year, graph the function for the domain, and maximize this graph. Second Year Formula: Part 5.Write a function for the total money earned over time using x as the percent raise for the first year, graph the function for the domain, and maximize this graph. Total Money Earned Formula: Part 6. What is the Range for the situation? How much of a difference does asking for the best combination versus the worst combination make for the starting salary? Part 8. On a separate paper, create a persuasive speech/pamphlet to the members of your workers union intended to influence them to pick the combination you feel is best.

8 22. When you cough, your windpipe contracts. The speed, v, with which air comes out depends on the radius, r, of your windpipe. If R is the normal (rest) radius of your windpipe, then for r R, the speed is given by: 2 v = a( R r) r where a is a positive constant. Part 1: What is the domain of r? Part 2: If R = 6 mm and a = 3, then sketch the graph over the domain. Part 3: What value of r maximizes the speed? Part 4: If R = 15 and a = 3, then sketch The graph over the domain. Part 5: What value of r maximizes the speed? Part 6: There is a pattern to how much the windpipe should contract to maximize speed. Analyze the starting and ending windpipe radii use percent, ratios, etc. and make a claim about this pattern.

9 23. A drug is being developed to raise the temperature of a person suffering from hypothermia. The temperature change, T (in Fahrenheit), in a patient generated by a dose, D (in grams), of a drug is given by " T = 3 2D % $ 'D 2. # 3 & Part 1: Given that the lethal dose of this drug is 4.5 grams, what is the domain? Part 2: Sketch the temperature change based on the dose. Part 3: What dose should be applied to a person who has A temp of 93.4 to raise the temperature to 98.6? Part 4: What dosage maximizes the temperature change? How would this affect a person with temperature 92? Part 5: If the ideal dose raises the temperature from 1 to 4 degrees, then what dosages will accomplish this goal?

10 Optimization Day The table below shows Municipal Solid Waste Disposal for a city: Year X Annual Landfill Disposal (millions of tons) Per Person Per Day (pounds) Part 1: Let x represent time in years since 1960 (so x = 0 means 1960). Fill in the x values above for the given years. Part 2: Let y represent the annual landfill disposal. Use regression analysis on a graphing utility to find a quadratic function that models this data. Part 3: If the landfill disposal continues to follow this trend, then when (to the nearest year) would the annual landfill disposal return to the 1970 level? Part 4: Is it reasonable to expect the annual landfill disposal to follow this trend indefinitely? Explain. Part 5: Let y represent the Per Person Per Day. Use regression analysis on a graphing utility to find a quadratic function that models this data. Part 6: If the landfill disposal continues to follow this trend, then when (to the nearest year) would the per person per day rate reduce to 1 pound per day? Part 7: Is it reasonable to expect the annual landfill disposal to follow this trend indefinitely? Explain.

11 25. The arch of a bridge is in the shape of a parabola 14 feet high at the center and 10 feet wide at the base. Part 1: Express the height of the arch h(x) in terms of x and state its domain. Part 2: Can a truck that is 8 feet wide and 12 feet high pass through the arch? Use the grid below and the table function of your calculator to sketch the heights of the bridge along the domain. Part 3: What is the tallest 8 foot wide truck that can pass through the arch? Use the grid above to verify your answer. Part 4: What is the widest 12 foot high truck that can pass through the arch? Don t just give a simple Integer answer be exact.

12 26. The roadbed of one section of a suspension bridge is hanging from a large cable suspended between two towers that are 200 feet apart. The cable forms a parabola that is 60 feet above the roadbed at the towers and 10 feet above the roadbed at the lower point. Part 1: Express the vertical distance d(x) (in feet) from the roadbed to the suspension cable in terms of x and state the domain of d(x). Part 2: The roadbed is supported by seven equally spaced vertical cables. Sketch these cables along a perfectly drawn parabola on the grid below. Part 3: What is the combined total length of these supporting cables?

13 Optimization Day Suppose you want to find the dimensions of the lightest cylindrical can containing ml. Part 1. Draw the can as a flat aluminum shape that would fold into the final can. Also draw the final can. Label the radius and height. Flat Can Final Can Part 2: Write a formula for the volume of the can and solve for the height, h. Part 3: Write a formula for the Surface Area where the only variable is the radius, r. Part 4: What is the domain of the Surface Area formula? Part 5: Write a formula for the weight of the can using the variable k for the weight of the material of the can if the top and bottom are made of a material that is three times as heavy (per unit area) as the material used for the side. (continued on back).

14 Part 6: Sketch the weight of the can on your stated domain. Part 7: What dimensions will minimize the amount of material (weight) needed to create the can if the value of k =.1 grams per square cm? Optimal: r = h = Weight = Part 8: How much will the can weigh when full of the liquid (1 liter = 1000 cubic cm = 1000 grams)? Part 9: What are the actual dimensions of your can? Part 10: State the error between your calculated optimal radius and the actual radius of the can. Part 11: State the error between your calculated optimal height and the actual height of the can. Group Evaluation: State the names of you group members and give them all a score out of 10 for their contribution to this project.