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4.6 Section http://www.webassign.net/v4cgijrmace2@ncsu/student.pl?h=200811... WebAssign Section 4.6 (Homework) MACE JOHN RANDALL MA 141, section 007, Fall 2008 Instructor: Kyle Thompson Current Score: 22 out of 22 Due: Thursday, October 30, 2008 10:04 PM EDT Description Optimization Problems The due date for this assignment is past. Your work can be viewed below, but no changes can be made. 1. [SCalcCC2 4.6.02.] 2/2 points Find two numbers whose difference is 122 and whose product is a minimum. -61 -61 (smaller number) 61 61 (larger number) 2. [SCalcCC2 4.6.04.] 1/1 points Find a positive number such that the sum of the number and its reciprocal is as small as possible. 1 1 3. [SCalcCC2 4.6.06.] 2/2 points Find the dimensions of a rectangle with area 1000 m2 whose perimeter is as small as possible. 31.622 31.6 m (length) 31.622 31.6 m (width) 4. [SCalcCC2 4.6.08.] 3/3 points Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and 1 of 3 PDF created with pdfFactory Pro trial version www.pdffactory.com 11/20/2008 10:20 PM Section 4.6 http://www.webassign.net/v4cgijrmace2@ncsu/student.pl?h=200811... bending up the sides. Find the largest volume that such a box can have. Let x denote the length of the side of the square being cut out. Let y denote the length of the base. (a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes. (Do this on paper. Your teacher may ask you to turn in this work.) (b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (Do this on paper. Your teacher may ask you to turn in this work.) (c) Write an expression for the volume V. V = xy^2 x*y^2 (d) Use the given information to write an equation that relates the variables. (Do this on paper. Your teacher may ask you to turn in this (e) work.) Use part (d) to write the volume as a function of x. V(x) = 4*x^3-12x^2+9*x (f) Finish solving the problem. V= 2 2 ft3 x*(3-2*x)^2 5. [SCalcCC2 4.6.10.] 2/2 points A box with a square base and open top must have a volume of 4000 cm3. Find the dimensions of the box that minimize the amount of material used. 20 20 cm (length and width) 10 10 cm (height) 6. [SCalcCC2 4.6.12.] 1/1 points A rectangular storage container with an open top is to have a volume of 10 m3. The length of this base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container. $ 163.5 163.54 7. [SCalcCC2 4.6.14.] 2/2 points 2 of 3 PDF created with pdfFactory Pro trial version www.pdffactory.com 11/20/2008 10:20 PM Section 4.6 http://www.webassign.net/v4cgijrmace2@ncsu/student.pl?h=200811... Find the point on the parabola that is closest to the point (0,-3). x + y2 = 0 ( -1 -1 , -1 -1 ) 8. [SCalcCC2 4.6.16.] 2/2 points Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola y = 8 - x2. 3.27 3.27 units (shorter side) 5.33 5.33 units (longer side) 9. [SCalcCC2 4.6.18.] 1/1 points Find the area of the largest rectangle that can be inscribed in the ellipse below. 2*a*b 2*a*b 10. [SCalcCC2 4.6.22.] 1/1 points A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? 16.65 16.65 ft 11. [SCalcCC2 4.6.26.] 5/5 points A boat leaves a dock at 2:00 P.M. and travels due south at a speed of 20 km/h. Another boat has been heading due east at 15 km/h and reaches the same dock at 3:00 P.M. At what time were the two boats closest together? 2 2 :2 21 1 :3 36 6 P.M. 3 of 3 PDF created with pdfFactory Pro trial version www.pdffactory.com 11/20/2008 10:20 PM

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