936 CHAPTER 9 Systems and Matrices 60. Unknown Numbers Find two numbers whose ratio is 4 to 3 and are such that the sum of their squares is 100. 61. Triangle Dimensions The longest side of a right triangle is 13 m in length. One of the other sides is 7 m longer than the shortest side. Find the lengths of the two shorter sides of the triangle. 62. Triangle Dimensions The longest side of a right triangle is 29 ft in length. One of the other two sides is 1 ft longer than the shortest side. Find the lengths of the two shorter sides of the triangle. Answer each question. 63. Does the straight line 3x - 2y = 9 intersect the circle x2 + y2 = 25? (Hint: To find out, solve the system formed by these two equations.) 64. For what value(s) of b will the line x + 2y = b touch the circle x2 + y2 = 9 in only one point? x y 0 66. Suppose we are given the equations of two circles that are known to intersect in exactly two points. How would we find the equation of the only chord common to these circles? –8 –4 4 0 8 x y y = x2 x2 + y2 = 90 –8 –4 8 65. A line passes through the points of intersection of the graphs of y = x2 and x2 + y2 = 90. What is the equation of this line? x x x x y y y Solve each problem. 67. Dimensions of a Box A box with an open top has a square base and four sides of equal height. The volume of the box is 360 ft3. If the surface area is 276 ft2, find the dimensions of the box. (Round answers to the nearest thousandth, if necessary.) 68. Dimensions of a Cylinder Find the radius and height (to the nearest thousandth) of an open-ended cylinder with volume 50 in.3 and lateral surface area 65 in.2. 2pr h h r 69. Checking Answers Check the two answers in Example 6. 70. (Modeling) Equilibrium Demand and Price The supply and demand equations for a certain commodity are given. supply: p = 2000 2000 - q and demand: p = 7000 - 3q 2q (a) Find the equilibrium demand. (b) Find the equilibrium price (in dollars).
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