If p = 4r^3 - 3r^2 + 6r - 2 and dr/dt = 3, find dp/dt at the instant r = 2. Air

Question

If p = 4r^3 - 3r^2 + 6r - 2 and dr/dt = 3, find dp/dt at the instant r = 2. Air is being pumped into a spherical balloon at the rate of 2 ft^3/min. What is the rate of change of the radius at the instant the radius of the balloon is 2ft? The volume of a sphere is V = 4/3 pi r^3. For each problem below. Draw a diagram. Label every quantity that changes with time using a variable and every quantity that doesn't change with a constant. Do not label anything with both a variable and a constant. Then, using the variables you have chosen, specify what is given in the problem and what you are trying to find (do not use any variable that does not appear on your diagram). Find a relationship between the quantities whose derivatives you are trying to relate. Finally, differentiate this relationship and solve the problem. a) A plane at an altitude of 2 miles is moving horizontally at a speed of 500 mi/hr away from an observer on the ground. At what rate is the distance between the plane and the observer changing at the instant the plane is a horizontal distance of 4 miles? b) A rocket that is launched vertically is tracked by a radar station located on the ground at a horizontal distance of 4 miles from the launch site. What is the vertical speed of the rocket the instant the in-air distance from the radar station is 5 miles and this distance is changing at the rate of 3600 mi/hr? For each part below, find the absolute maximum and absolute minimum of the given function over the stated interval. a) f(x) = x^3 - 3x^2 - 9x + 15, [-2, 4] b) f(x) = 1 - x/x^2 + 3, [-2, 5]

Explanation / Answer

1 . p = 4r3-3r2+6r-2

dp/dt = (12r2-6r+6)(dr/dt)

dp/dt=(12(2)2-6(2)+6)(3)

dp/dt = (48 -12 +6)(3)

dp/dt = 126

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