830 CHAPTER 11 Systems of Equations and Inequalities In Problems 90–96, use Descartes’ method from Problem 89 to find an equation of the tangent line to each graph at the given point. 90. + = x y 10; 2 2 at ( ) 1, 3 91. = + y x 2; 2 at ( ) 1, 3 92. + = x y 5; 2 at ( ) −2, 1 93. + = x y 2 3 14; 2 2 at ( ) 1, 2 94. + = x y 3 7; 2 2 at ( ) −1, 2 95. − = x y 3; 2 2 at ( ) 2, 1 96. − = y x 2 14; 2 2 at ( ) 2, 3 97. If r1 and r2 are two solutions of a quadratic equation + + = ax bx c 0, 2 it can be shown that + = − = r r b a r r c a and 1 2 1 2 Solve this system of equations for r1 and r .2 98. Challenge Problem Solve for x and y in terms of ≠ a 0 and ≠ b 0: + = + + = + ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪ x a y b a b a b x a y b a b ab 2 2 2 2 2 2 2 2 99. Challenge Problem Geometry Find formulas for the length l and width w of a rectangle in terms of its area A and perimeter P. 100. Challenge Problem Geometry Find formulas for the base b and one of the equal sides l of an isosceles triangle in terms of its altitude h and perimeter P. 1 01. A circle and a line intersect at most twice. A circle and a parabola intersect at most four times. Deduce that a circle and the graph of a polynomial of degree 3 intersect at most six times. What do you conjecture about a polynomial of degree 4? What about a polynomial of degree n? Can you explain your conclusions using an algebraic argument? 102. Suppose you are the manager of a sheet metal shop. A customer asks you to manufacture 10,000 boxes, each box being open on top. The boxes are required to have a square base and a 9-cubic-foot capacity. You construct the boxes by cutting out a square from each corner of a square piece of sheet metal and folding along the edges. (a) Find the dimensions of the square to be cut if the area of the square piece of sheet metal is 100 square feet. (b) Could you make the box using a smaller piece of sheet metal? Make a list of the dimensions of the box for various pieces of sheet metal. Explaining Concepts 103. Solve: = − x x 7 8 6 2 104. Find an equation of the line with slope − 2 5 that contains the point ( ) − 10, 7. 105. If θ = cot 24 7 and θ < cos 0, find the exact value of each of the remaining trigonometric functions. 106. Finding the Grade of a Mountain Trail A straight trail with uniform inclination leads from a hotel, elevation 5300 feet, to a lake in the valley, elevation 4100 feet. The length of the trail is 4420 feet. What is the inclination (grade) of the trail? 107. Find an equation of the circle with center at ( ) −3, 4 and radius 10. 108. Solve: < + x x4 21 2 109. Find the function that is finally graphed after = − y x 25 2 is reflected about the x-axis and shifted right 4 units. 110. If ( ) = − + f x x x 2 8 7, 2 find ( ) − f x 3 . 111. Find the difference quotient of ( ) = − f x x x 3 8 . Simplify the answer. 112. Simplify: [ ] ( ) ( ) ( ) − ⋅ − ⋅ − ⋅ − x x x x 2 5 3 3 9 2 5 2 2 5 9 8 9 2 Retain Your Knowledge Problems 103–112 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus.
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