Problem 1. (1 point) Problem 6. (1 point) Suppose xy = -1 and dy - = 2. Find - when x = 1. Helium is pumped into a spherical balloon at a rate of 5 cubic feet dt dt per second. How fast is the radius increasing after 2 minutes? dx Note: The volume of a sphere is given by V = (4/3)nur. Problem 2. (1 point) Rate of change of radius (in feet per second) = If x2 + y2 = 2, and =5 when x = 1 and y = 1, what is dy dt - when dt Problem 7. (1 point) x = 1 and y = 1? A street light is at the top of a 12-ft-tall pole. A 6-ft tall woman walks away from the pole with a speed of 8 ft/sec along a straight dy path (see figure). How fast is the tip of her shadow moving, in dt ft/sec, when she is 45 ft from the base of the pole? Problem 3. (1 point) dt If y2 + xy - 3x= -9, and = -2 when x = 3 and y = -3, what dx when x = 3 and y = -3? dx dt Problem 4. (1 point) A particle is moving along the curve y = 5v4x + 5. As the parti- Problem 8. (1 point) cle passes through the point (1, 15), its x-coordinate increases at a Water is leaking out of an inverted conical tank at a rate of 10200.0 rate of 5 units per second. Find the rate of change of the distance cubic centimeters per min at the same time that water is being from the particle to the origin at this instant. pumped into the tank at a constant rate. The tank has height 6.0 meters and the diameter at the top is 4.5 meters. If the wa- Problem 5. (1 point) ter level is rising at a rate of 19.0 centimeters per minute when The radius of a spherical balloon is increasing at a rate of 3 cen- the height of the water is 4.5 meters, find the rate at which wa- timeters per minute. How fast is the volume changing, in cubic ter is being pumped into the tank in cubic centimeters per minute. centimeters per minute, when the radius is 12 centimeters? Note: The volume of a sphere is given by V = (4/3) 13. Note: Let "R" be the unknown rate at which water is being pumped in. Then you know that if V is volume of water, d, = Rate of change of volume, in cubic centimeters per minute, = R - 10200.0. Use geometry (similar triangles?) to find the rela- tionship between the height of the water and the volume of the water at any given time. Recall that the volume of a cone with base radius r and height h is given by anr h