270 Chapter 4 Applications of Derivatives c. What is the largest area the rectangle can have, and what are its dimensions? 12. Find the volume of the largest right circular cone that can be in- X scribed in a sphere of radius 3. NB P(x, ? ) A , x 13. Two sides of a triangle have lengths a and b, and the angle be- ween them is 0. What value of 0 will maximize the triangle's 4. A rectangle has its base on the x-axis and its upper two vertices on area? (Hint: A = (1/2)absin 0.) the parabola y = 12 - x. What is the largest area the rectangle can have, and what are its dimensions? 14. Designing a can What are the dimensions of the lightest open-top right circular cylindrical can that will hold a volume of 5. You are planning to make an open rectangular box from an 1000 cm ? Compare the result here with the result in Example 2. 8-in.-by-15-in. piece of cardboard by cutting congruent squares 15. Designing a can You are designing a 1000 cm' right circular from the corners and folding up the sides. What are the dimen- cylindrical can whose manufacture will take waste into account. sions of the box of largest volume you can make this way, and There is no waste in cutting the aluminum for the side, but the top what is its volume? and bottom of radius r will be cut from squares that measure 2r 6. You are planning to close off a comer of the first quadrant with a units on a side. The total amount of aluminum used up by the can line segment 20 units long running from (a, 0) to (0, b). Show that will therefore be the area of the triangle enclosed by the segment is largest when A = 8r2+ 2Trrh a = b. rather than the A = 2mr2 + 21rrh in Example 2. In Example 2, 7. The best fencing plan A rectangular plot of farmland will be the ratio of h to r for the most economical can was 2 to 1. What is bounded on one side by a river and on the other three sides by a the ratio now? single-strand electric fence. With 800 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions? T 16. Designing a box with a lid A piece of cardboard measures 0 in. by 15 in. Two equal squares are removed from the corners 8. The shortest fence A 216 m' rectangular pea patch is to be of a 10-in. side as shown in the figure. Two equal rectangles are enclosed by a fence and divided into two equal parts by another removed from the other corners so that the tabs can be folded to fence parallel to one of the sides. What dimensions for the outer form a rectangular box with lid. rectangle will require the smallest total length of fence? How much fence will be needed? HX+ 9. Designing a tank Your iron works has contracted to design and build a 500 fts, square-based, open-top, rectangular steel holding NOT TO SCALE tank for a paper company. The tank is to be made by welding thin 10' Base Lid 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. 15- a. What dimensions do you tell the shop to use? b. Briefly describe how you took weight into account. a. Write a formula V(x) for the volume of the box. 10. Catching rainwater A 1125 fts open-top rectangular tank with b. Find the domain of V for the problem situation and graph V square base x ft on a side and y ft deep is to be built with its top over this domain. dush with the ground to catch runoff water. The costs associated c. Use a graphical method to find the maximum volume and the with the tank involve not only the material from which the tank is value of x that gives it. made but also an excavation charge proportional to the product xy. d. Confirm your result in part (c) analytically. a. If the total cost is T 17. Designing a suitcase A 24-in.-by-36-in. sheet of cardboard is folded in half to form a 24-in.-by-18-in. rectangle as shown in the ac- c = 5(x2 + 4xy) + 10xy, companying figure. Then four congruent squares of side length x are what values of x and y will minimize it ? cut from the corners of the folded rectangle. The sheet is unfolded, b. Give a possible scenario for the cost function in part (a). and the six tabs are folded up to form a box with sides and a lid. 11. Designing a poster You are designing a rectangular poster to a. Write a formula V(x) for the volume of the box. contain 50 in of printing with a 4-in. margin at the top and bot- b. Find the domain of V for the problem situation and graph V tom and a 2-in. margin at each side, What overall dimensions will over this domain. minimize the amount of paper used ?4.6 Applied Optimization 271 c. Use a graphical method to find the maximum volume and the 21. (Continuation of Exercise 20.) value of x that gives it. a. Suppose that instead of having a box with square ends you d. Confirm your result in part (c) analytically. have a box with square sides so that its dimensions are h by . Find a value of x that yields a volume of 1120 in'. h by w and the girth is 2h + 2w. What dimensions will give the box its largest volume now? f. Write a paragraph describing the issues that arise in part (b). Girth 24" 24" -36" 18" The sheet is then unfolded. T b. Graph the volume as a function of h and compare what you see with your answer in part (a). 22. A window is in the form of a rectangle surmounted by a semi- circle. The rectangle is of clear glass, whereas the semicircle is of 24" Base tinted glass that transmits only half as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the thickness of the frame. -36 18. A rectangle is to be inscribed under the arch of the curve y = 4 cos (0.5x) from x = -T to x = 1. What are the dimen- sions of the rectangle with largest area, and what is the largest area? 19. Find the dimensions of a right circular cylinder of maximum vol- ume that can be inscribed in a sphere of radius 10 cm. What is the maximum volume? 23. A silo (base not included) is to be constructed in the form of a 20. a. The U.S. Postal Service will accept a box for domestic ship- cylinder surmounted by a hemisphere. The cost of construction ment only if the sum of its length and girth (distance around) per square unit of surface area is twice as great for the hemisphere does not exceed 108 in. What dimensions will give a box with as it is for the cylindrical sidewall. Determine the dimensions to a square end the largest possible volume? be used if the volume is fixed and the cost of construction is to be Girth = distance kept to a minimum. Neglect the thickness of the silo and waste in around here construction. 24. The trough in the figure is to be made to the dimensions shown Only the angle 0 can be varied. What value of 0 will maximize trough's volume? Length Square end T b. Graph the volume of a 108-in. box (length plus girth equals 108 in.) as a function of its length and compare what you see with your answer in part (a).272 Chapter 4 Applications of Derivatives 25. Paper folding A rectangular sheet of 8.5-in.-by-1 1-in. paper is placed on a flat surface. One of the corners is placed on the oppo- 31. A wire b m long is cut into two pieces. One piece is bent into an site longer edge, as shown in the figure, and held there as the paper equilateral triangle and the other is bent into a circle. If the sum of is smoothed flat. The problem is to make the length of the crease the areas enclosed by each part is a minimum, what is the length as small as possible. Call the length L. Try it with paper. of each part? n. Show that 12 = 2x3/ (2x - 8.5). 32. Answer Exercise 31 if one piece is bent into a square and the other into a circle. . What value of x minimizes L? . What is the minimum value of L? 33. Determine the dimensions of the rectangle of largest area that can be inscribed in the right triangle shown in the accompanying D figure. 2 2 - x 2 L\\Crease Q (originally at A) 34. Determine the dimensions of the rect- angle of largest area that can be inscribed in a semicircle of radius 3. (See accompanying figure.) 26. Constructing cylinders Compare the answers to the following = 3 two construction problems. 35. What value of a makes f(x) = x2 + (a/x) have a. A rectangular sheet of perimeter 36 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of a. a local minimum at x = 2? the figure. What values of x and y give the largest volume? b. a point of inflection at x = 1? b. The same sheet is to be revolved about one of the sides of 36. What values of a and b make f(x) = x3 + ax + bx have length y to sweep out the cylinder as shown in part (b) of the a. a local maximum at x = - 1 and a local minimum at x = 3? figure. What values of x and y give the largest volume? b. a local minimum at x = 4 and a point of inflection at x = 1? 37. A right circular cone is circumscribed in a sphere of radius 1. De- termine the height h and radius r of the cone of maximum volume. 38. Determine the dimensions of the inscribed rectangle of maximum area. Circumference = x y = ex 27. Constructing cones A right triangle whose hypotenuse is V3 m long is revolved about one of its legs to generate a right circular cone. Find the radius, height, and volume of the cone of 39. Consider the accompanying graphs of y = 2x + 3 and y = Inx. greatest volume that can be made this way. Determine the a. minimum vertical distance b. minimum horizontal distance between these graphs. 1y = 2x + 3 28. Find the point on the line a + b + = 1 that is closest to the origin. 3 y = In x 29. Find a positive number for which the sum of it and its reciprocal is KNIW he smallest (least) possible. -> x 30. Find a positive number for which the sum of its reciprocal and 1 four times its square is the smallest possible