5.8E: Modeling Using Variation (Exercises)
For the following exercises, find the unknown value.
43. \(y\) varies directly as the square of \(x\). If when \(x=3, \quad y=36,\) find \(y\) if \(x=4\).
44. \(y\) varies inversely as the square root of \(x\) If when \(x=25, \quad y=2,\) find \(y\) if \(x=4\).
45. \(y\) varies jointly as the cube of \(x\) and as \(z\). If when \(x=1\) and \(z=2, y=6,\) find \(y\) if \(x=2\) and \(z=3\).
46. \(y\) varies jointly as \(x\) and the square of \(z\) and inversely as the cube of \(w\). If when \(x=3, z=4\), and \(w=2, y=48\), find \(y\) if \(x=4, z=5,\) and \(w=3 .\)
For the following exercises, solve the application problem.
47. The weight of an object above the surface of the earth varies inversely with the distance from the center of the earth. If a person weighs 150 pounds when he is on the surface of the earth \((3,960\) miles from center), find the weight of the person if he is 20 miles above the surface.
48. The volume \(V\) of an ideal gas varies directly with the temperature \(T\) and inversely with the pressure P. A cylinder contains oxygen at a temperature of 310 degrees \(\mathrm{K}\) and a pressure of 18 atmospheres in a volume of 120 liters. Find the pressure if the volume is decreased to 100 liters and the temperature is increased to 320 degrees \(K\).