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Suppose that a dimension x and the area A = 12x of a shape are differentiable functions of t. Write an equation that dA dx relates to dt dt . . . dA dx dt dtdx Assume that x = x(t) and y = y(t). Let y = x + 3 and =2 when x = 1. dt Find when x = 1. dt . . . dy dt (Simplify your answer.)The original 24 m edge length x of a cube decreases at the rate of 5 m/min. a. When x = 5 m, at what rate does the cube's surface area change? b. When x = 5 m, at what rate does the cube's volume change? The dimensions x and y of an object are related to its volume V by the formula V = 12x y. dV a. How is dy dt related to if x is constant? dt dV dx b. How is related to- dt dt if y is constant? dV dx dy c. How is dt related to dt and dt if neither x nor y is constant?When a circular plate of metal is heated in an oven, its radius increases at a rate of 0.02 cm/min. At what rate is the plate's area increasing when the radius is 50 cm? @ Write an equation relating the area of the circular plate, A, and the radius, r. E A child ies a kite at a height of 90 it, the wind carrying the kite horizontally away from the child at a rate of 30 ft/sec. How fast must the child let out the string when the kite is 150 ft away from the child? Sand falls from a conveyor belt at a rate of 9 m3 l min onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. How fast are the height and the radius changing when the pile is 5 m high? Answer in centimeters per minute. Suppose that a drop of mist is a perfect sphere and that, through condensation, the drop picks up moisture at a rate proportional to its surface area. Show that under these circumstances the drop's radius increases at a constant rate. A spherical iron ball 12 in. in diameter is coated with a layer of ice of uniform thickness. If the ice melts at a rate of 13 in.3/min. how fast is the thickness of the ice decreasing when it is 5 in. thick? How fast is the outer surface area of ice decreasing? A balloon is rising vertically above a level, straight road at a constant rate of 3 ft/ sec. Just when the balloon is 70 ft above the ground, a bicycle moving at a constant rate of 11 ft/ sec passes under it. How fast is the distance s(t) between the bicycle and balloon increasing 6 seconds later? 5' s(t) L .L.:': [I lit} dx The coordinates ofa particle in the metric xy-plane are differentiable functions of time t with a = - 5 m/sec and dy . . . . . . . . a = - 8 m/sec. How fast Is the particle's distance from the origin changing as It passes through the pomt (12,5)?