Related Rates Multiple choice 1. Let x and y be functions of time t such that the sum of x and y is constant. Which of the following equations describes the relationship between the rate of change of x with respect to time and the rate of change of y with respect to time? a) dx/dt = dy/dt b) dx/dt = -dy/dt c) dx/dt + dy/dt = dK/dt, where K is a function of t d) dx/dt + dy/dt = K, where K is a function of t 2. Boyle's law states that if the temperature of an ideal gas is held constant, then the pressure P of the gas and its volume V satisfy the relationship P= k/V , where k is a constant. Which of the following best describes the relationship between the rate of change, with respect to time t , of the pressure and the rate of change, with respect to time t, of the volume? a) dP/dt = k/(dV/dt) b) dP/dt = -k/(dV/dt) c) dP/dt = (k/v^2) (dV/dt) d) dP/dt = (-k/v^2) (dV/dt) 3. A right triangle has base x feet and height h feet, where x is constant and h changes with respect to time t, measured in seconds. The angle , measured in radians, is defined by tan= h/x. Which of the following best describes the relationship between d/dt, the rate of change of with respect to time, and dh/dt, the rate of change of h with respect to time? a) d/dt = (x/(x^2+h^2))dh/dt radians per second b) d/dt = (x^2/(x^2+h^2)) dh/dt radians per second c) d/dt = (1/( x^2+h^2)) dh/dt radians per second d) d/dt = tan^1((1/x) dh/dt) radians per second 4. Let x and y be functions of time t such that the sum of x and twice y is constant. Which of the following equations describes the relationship between the rate of change of x with respect to time and the rate of change of y with respect to time? a) dx/dt = 2dy/dt b) dx/dt = -2dy/dt c) 2dx/dt + dy/dt = 0 d) dx/dt + 2 dy/dt = K, where K is a function of t 5. A right triangle has base x meters and height h meters, where h is constant and x changes with respect to time t , measured in seconds. The angle , measured in radians, is defined by tan= h/x. Which of the following best describes the relationship between d/dt, the rate of change of with respect to time, and dx/dt, the rate of change of x with respect to time? a) d/dt = (-h/(x^2+h^2)) dx/dt radians per second b) d/dt = (x/(x^2+h^2)) dx/dt radians per second c) d/dt = (-h/(x x^2+h^2)) dx/dt radians per second d) d/dt = (h/(x x^2+h^2)) dx/dt radians per second 6. In a certain factory, assume that the number of workers is constant. The number of minutes N that it takes to make a single unit of a product and the number of units U of the product that are made per day satisfy the relationship U=k/N, where k is a constant. Which of the following best describes the relationship between the rate of change, with respect to time t, of U and the rate of change, with respect to time t, of N? a) dU/dt = k/((dN/dt)) b) dU/dt = -k/((dN/dt)) c) dU/dt = (k/N^2) (dN/dt) d) dN/dt = (-k/N^2) (dN/dt)