MCV 4UOptimization Problems AssignmentName:
1.A manufacturer of microwave ovens will, on average, sell 300 units a month at $400 per unit. It has been determined that the company can sell an additional 100 ovens for each reduction of $20 in price.
Determine the price that will maximize revenue. [$230]
2.A cable company is laying cable in an area with underground utilities. Two subdivisions are located on opposite sides of Willow Creek, which is 100 m wide. The company has to connect points P and Q with cable, where Q is on the north bank 1200 m east of P. It costs $40/m to lay cable underground and $80/m to lay cable underwater. What is the least expensive way to lay the cable?
[underground to a point 1142 m east of P, lay remaining cable underwater.]
3.If 2400 cm2 of material is available to construct an open-topped box that is to have a square base, find the dimensions that create the box of maximum volume. []
4.A straight section of railroad track crosses two highways at points that are 400 m and 600 m, respectively, from an intersection. (As illustrated below) Determine the dimensions of the largest rectangular lot that can be laid out in the triangle formed by the tracks and highways. [300 m X 200 m]
5.Sarah leaves her house at 9:00 and travels due east at a speed of 80 km/h. Adam has been heading south at 70 km/h and reaches Sarah’s house at 10:00. When were the cars closest together? [9:26]
6.The Bouchard Soup Company estimates that the cost, ($), of making x cans of chicken noodle soup is
And the revenue is
a)Find the average cost function.
b)At what production level will the average cost be smallest? [15 492 cans]
c)In order to maximize profits, how many cans of soup should be sold? [10 000 cans]
7.The lifeguard at a public beach has 400 m of rope available to lay out a rectangular restricted swimming area using the straight shoreline as one side of the rectangle. She wants to maximize the swimming area. What will the dimensions of the rectangle be? [200mX100m]
8.A square piece of cardboard with side 60 cm is made into an open rectangular box by cutting equal squares from each corner and folding up the sides.
a)Determine the length of the sides of the squares if the volume of the box is a maximum. [10 cm]
b)What is the volume of the box? [16 000 cm3]
9.A Norman window has the shape of a rectangle capped by a semicircular region as shown. If the perimeter of the window is 8 m, find the width of the window that will permit the greatest amount of light.
[1.12 m]
10.A boat leaves a dock at noon and heads west at a speed of 25 km/h. Another boat heads north at 20 km/h and reaches the same dock at 1:00 pm. When were the boats closest to each other? [12:23]
11.The position, at time t, in seconds, of an object moving along a line is given by
s(t) = 3t3 – 40.5t2 + 162t,
a) Determine the velocity and the acceleration at any given time, t.
b) When if the object stationary? When is it moving in a positive direction? When is it moving in a negative direction?
c) At what time, t, is the object “at rest”?
d) At what time, t, is the object slowing down?
e) At what time, t, is the speeding up?
12.Determine the maximum and minimum values of .
13.A wall is 1.8 m high and 1.2 m from a building. Find the length of the shortest ladder that will touch the building, the top of the wall, and the ground beyond the wall. {4.2 m}
Assignment: Minimum to be completed for Friday, May 16th # 1, 2, 4, 5, 9, 11, 13
Before the end of the unit you should complete all remaining questions.