136 CHAPTER 5. THE DERIVATIVE

15. The ordered pair (x,y) is on the ellipse x2 +4y2 = 4. Form the rectangle which has(x,y) as one end of a diagonal and (0,0) at the other end. Find the rectangle of thissort which has the largest possible area.

16. A rectangle is inscribed in a circle of radius r. Find the formula for the rectangle ofthis sort which has the largest possible area.

17. A point is picked on the ellipse x2 + 4y2 = 4 which is in the first quadrant. Then aline tangent to this point is drawn which intersects the x axis at a point x1 and the yaxis at the point y1. The area of the triangle formed by the y axis, the x axis, and theline just drawn is thus x1y1

2 . Out of all possible triangles formed in this way, find theone with smallest area.

18. Find maximum and minimum values if they exist for the function f (x) = lnxx for

x > 0.

19. Describe how you would find the maximum value of the function f (x) = lnx2+sinx for

x ∈ (0,6) if it exists. Hint: You might want to use a calculator to graph this and getan idea what is going on.

20. A rectangular beam of height h and width w is to be sawed from a circular log ofradius 1 foot. Find the dimensions of the strongest such beam assuming the strengthis of the form kh2w. Here k is some constant which depends on the type of woodused.

w

h1

21. A farmer has 600 feet of fence with which to enclose a rectangular piece of land thatborders a river. If he can use the river as one side, what is the largest area that he canenclose.

22. An open box is to be made by cutting out little squares at the corners of a rectangularpiece of cardboard which is 20 inches wide and 40 inches long and then folding upthe rectangular tabs which result. What is the largest possible volume which can beobtained?

23. A feeding trough is to be made from a rectangular piece of metal which is 3 feet wideand 12 feet long by folding up two rectangular pieces of dimension one foot by 12feet. What is the best angle for this fold?

24. Find the dimensions of the right circular cone which has the smallest area given thevolume is 30π cubic inches. The volume of the right circular cone is (1/3)πr2h andthe area of the cone is πr

√h2 + r2.

25. A wire of length 10 inches is cut into two pieces, one of length x and the other oflength 10−x. One piece is bent into the shape of a square and the other piece is bentinto the shape of a circle. Find the two lengths such that the sum of the areas of thecircle and the square is as large as possible. What are the lengths if the sum of thetwo areas is to be as small as possible.