The top of a silo has the shape of a hemisphere of diameter 20 ft. If it is coated uniformly with a layer of ice and if the thickness is decreaing at a rate of 1/4 in/hr, how fast is the volume of the ice changing when the ice is 2 inches thick?A ship is steaming due north at 12 knots (1 knot = 1.85 kilometers/hour) and sights a large tanker 3 kilometers away northwest steaming at 15 knots due east. For safety reasons, the ship wants to maintain a distance of atleast 100 meters between them. Determine the shortest distance between the ships to determine if they can remain on their current headings or need to change course.A water tank has the shape of an inverted right circularcone of altitude 12 feet and base radius 6 feet. If water is being pumpedinto the tank at a rate of 10 gal/min. Approximate the rate at which the water level is rising when the water is 3 feet deep (1 gal= .1337ft^3).These three have stumped me. Any ideas would be appreciated! Thanks!!
1. volume of hemisphere is V = 2pi/3 r^3 dV/dt = 2pi r^2 dr/dt = 2pi * 2^2 * 1/4 = 2pi in^3/hr (volume change rate) 2. let the intersecting point of their current path be O(0,0), the two ships current coordinate is (0,3000/sqrt(2)) and (-3000/sqrt(2),0) assuming the ship due north has a coordinate of y, and the other x, y = 12 * 1850 t - 3000/sqrt(2) = 22200 t - 3000/sqrt(2) (in m /hr), x = 15 * 1850 t - 3000/sqrt(2) = 27750 t - (3000/sqrt(2)) t is the time from the current moment the distance between the two is d = sqrt(x^2 + y^2) which gives you a function of t take the derivative of d over t and find its minimum, if the minimum is greater than 100, it's safe it can be found that the shortest distance between the two ships is 5 minutes later when they are about 330 m apart so it's safe
