Applied Optimization Problems Homework

Directions: Write legibly and in pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution, show all steps, and write the final answer in a sentence. Do not turn in your scratch work. Staple your pages together, in the correct order, and use this page as a cover sheet.

(1) You are planning to close off a corner of the first quadrant with a line segment 20 units long running from to Show that the area of the triangle enclosed by the segment is largest when

(2) Your iron work has contracted to design and build a square-based, open-top, rectangular steel holding tank for a paper company. The tank is to be made by welding thin stainless steel plates together along their edges. As the production engineer, your job is to find dimensions for the base and height that will make the tank weigh as little as possible. (a) What dimensions do you tell the shop to use? (b) Briefly describe how you took weight into account.

(3) The bottom of an 8-ft-high mural painted on a vertical wall is 13 ft above the ground. The lens of a camera fixed to a tripod is 4 ft above the ground. How far from the wall should the camera be placed to photograph the mural with the largest possible angle?

(4) A 1125 open-top rectangular tank with a square base ft on a side and ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product (a) If the total cost is what values of and will minimize it? (b) Give a possible scenario fro the cost function in part (a).

(5) Two sides of a triangle have lengths and , and the angle between then What value of will maximize the triangle's area?

(6) The height of an object moving vertically is given by with in feet and in seconds. Find the object's velocity when Find its maximum height and when it occurs. Also find its velocity when

(7) Find the dimensions of the right circular cylinder of largest volume that can be inscribed in a sphere of radius

(8) Jane is 2 mi offshore in a boat and wishes to reach a coastal village 6 mi down a straight shoreline from the point nearest the boat. She can now row 2 mph can walk 5 mph. Where should she land her boat to reach the village in the least amount of time?

(9) The positions of two particles on the -axis are and with and in meters and in seconds. (a) At what time(s) in the interval do the particles meet? (b) What is the farthest apart that the particles ever get? (c) When in the interval is the distance between the particles chaining the fastest?  

(10) A truck is 250 mi due east of a sports car and is traveling west at a constant speed of 60 mi/h. Meanwhile, the sports car is going north at 80 mi/h. When will the truck and the car be closest to each other? What is the minimum distance between them?

(11) A jeep is on the desert at a point located 40 km from a point , which lies on a long straight road. The driver can travel at 45 km/h on the desert and 75 km//h on the road. The driver will win a prize if he arrives at the finish line at point , 50 km from , in 84 minutes or less. What route should he travel to minimize the time of travel? Does he win the prize?

• print this page
• pages linking here
• related pages