SECTION 11.6 Systems of Nonlinear Equations 829 84. Running a Race In a 1-mile race, the winner crosses the finish line 10 feet ahead of the second-place runner and 20 feet ahead of the third-place runner. Assuming that each runner maintains a constant speed throughout the race, by how many feet does the second-place runner beat the third-place runner? meter per hour faster than the tortoise, crosses the finish line 3 minutes before the tortoise. What are the average speeds of the tortoise and the hare? 21 meters 85. Constructing a Box A rectangular piece of cardboard, whose area is 216 square centimeters, is made into an open box by cutting a 2-centimeter square from each corner and turning up the sides. See the figure. If the box is to have a volume of 224 cubic centimeters, what size cardboard should you start with? 86. Constructing a Cylindrical Tube A rectangular piece of cardboard, whose area is 216 square centimeters, is made into a cylindrical tube by joining together two sides of the rectangle. See the figure. If the tube is to have a volume of 224 cubic centimeters, what size cardboard should you start with? 87. Fencing A farmer has 300 feet of fence available to enclose a 4500-square-foot region in the shape of adjoining squares, with sides of length x and y. See the figure. Find x and y. y x x y 88. Bending Wire A wire 60 feet long is cut into two pieces. Is it possible to bend one piece into the shape of a square and the other into the shape of a circle so that the total area enclosed by the two pieces is 100 square feet? If this is possible, find the length of the side of the square and the radius of the circle. 89. Descartes’ Method of Equal Roots Descartes’ method for finding tangent lines depends on the idea that, for many graphs, the tangent line at a given point is the unique line that intersects the graph at that point only. We use his method to find an equation of the tangent line to the parabola = y x2 at the point ( ) 2, 4 . See the figure. x y y 5 x2 22 23 21 1 2 3 1 2 3 4 5 21 (2, 4) y 5 mx 1 b First, an equation of the tangent line can be written as = + y mx b. Using the fact that the point ( ) 2, 4 is on the line, we can solve for b in terms of m and get the equation ( ) = + − y mx m 4 2 . Now we want ( ) 2, 4 to be the unique solution to the system = = + − ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ y x y mx m 4 2 2 From this system, we get ( ) − + − = x mx m2 4 0. 2 Using the quadratic formula, we get ( ) = ± − − x m m m 4 2 4 2 2 To obtain a unique solution for x, the two roots must be equal; in other words, the discriminant ( ) − − m m 4 2 4 2 must be 0. Complete the work to get m, and write an equation of the tangent line.
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