CHAPTER 3, PRACTICE EXERCISE, 118JOHN D. MCCARTHYPoints moving on coordinate axes Points A and B move along the x− andy−axes, respectively, in such a way that the distance r (meters) along the perpen-dicular from the origin to the line AB remains constant. How fast is OA changing,and is it increasing, or decreasing, when OB = 2r and B is moving toward O atthe rate of 0.3r m/sec?Solution We shall assume, without any loss of generality, that A is on the positivex-axis and B is on the positive y-axis.Let a be the x-coordinate of A and y be the y-c oordinate of B. Then A = (a, 0),a > 0, B = (0, y), and y > 0. Let O be the origin. Then O = (0, 0). Let P be thefoot of the p erpendicular from O to line AB. Then r = OP and r is constant.Note that OA = a and OB = y. Moreover, P is strictly between A and B.Let θ be the radian measure of ∠BAO = ∠P AO, ψ be the radian measure of∠AOP , and α be the radian measure of ∠P OB. Then θ + ψ = π/2 = α + ψ and,hence, θ = α. Hence:(1) cos(θ) = cos(α) = cos(∠P OB) = r/OB = r/yand(2) sin(θ) = sin(∠P AO) = r/OA = r/a.By the Pythagorean Theorem, it follows that:(3) (r/y)2+ (r/a)2= cos2(θ) + sin2(θ) = 1and, hence:(4) a−2+ y−2= r−2.Differentiating both sides of equation (4) with respect to t, we obtain:(5) (−2)(a−3)(da/dt) + (−2)(y−3)(dy/dt) = 0and, hence:(6) y3(da/dt) + a3(dy/dt) = 0.Now suppose that OB = 2r and B is moving toward O at the rate of 0.3r m/sec.Then y = OB = 2r and dy/dt = −0.3r. Thus, from equation (4), it follows that:Date: February 24, 201 0.12 J. MCCARTHY(7) a−2+ (1/4)r−2= r−2and, hence, a−2= (3/4)r−2; that is to say, a2= (4/3)r2. Since a > 0, this impliesthat a = 2r/31/2.Since y = 2r, a = 2r/31/2, and dy/dt = −0.3r, it follows from equation (6) that:(8) (8r3)(da/dt) + (8r3/33/2)(−0.3r) = 0and, hence:(9) da/dt = 0.3/33/2= 0.1/31/2= 31/2/30.In conclusion, OA is changing at the rate of 31/2/30 m/sec when OB = 2r andB is moving toward O at the rate of 0.3r m/sec. Moreover, since 31/2/30 > 0, OAis increasing when OB = 2r and B is moving toward O at the rate of 0.3r m/sec.Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027E-mail address: [email protected]: