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Question 1)

A hot air balloon rising vertically is tracked by an observer located 5 miles from the lift-off point. At a certain moment, the angle between the observer's line-of-sight and the horizontal is ~ , and it is changing at a rate of 0.1 rad/min. How fast is the balloon rising at this moment? miles/minA jogger runs around a circular track of radius 65 ft. Let (x, y) be her coordinates, where the origin is the center of the track. When the jogger's coordinates are (39, 52), her x-coordinate is changing at a rate of 16 ft/s. Find dy/dt. dy/dt = ft/sA rope, attached to a weight, goes up through a pulley at the ceiling and back down to a worker. The worker holds the rope at the same height as the connection point between the rope and weight. The distance from the connection point to the ceiling is 25 ft. Suppose the worker stands directly next to the weight (Le, a total rope length of 50 ft) and begins to walk away at a constant rate of 4 ftfs. How fast is the weight rising when the worker has walked: 10 feet? Answer: 30 feet? Answer: (1 point)A street light is at the top of a 10.0 ft. tall pole. A man 5.5 ft tall walks away from the pole with a speed of 5.0 feeti'sec along a straight path. How fast is the tip of his shadow moving when he is 47 feet from the pole? Your answer: ft/sec Hint: Draw a picture and use similar triangles. (1 point) Gravel is being dumped from a conveyor belt at a rate of 50 ftS/min. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 12 ft high? The height is increasing at ftlmin. (1 point) For the function f given above, determine whether the following conditions are true. Input 7 if the condition is ture, otherwise input F . (a) f' (a) 0 if ac > 2; (c) f" (a) 0 if1 > 4; (f) Two inflection points of f () are, the smaller one is * = and the other is x =Find the intervals on which the function is concave up or down, the points of inflection, and the critical points, and determine whether each critical point corresponds to a local minimum or maximum (or neither). Let x) : 2xl2sin[:c),0 S x 5 21 What are the critical point(s) = What does the Second Derivative Test tell about the first critical point: ? v '? What does the Second Derivative Test tell about the second critical point: '? v ? What are the inflection Point(s) = On lntervali is f '? v ? is f' ? v ? On Interval 2 is f ? v '? is f' ? v '? On lntervali is f '? v '? On lnterval2 is f ? v '? (1 point) Suppose that it?) = (3'3 + 4W 1)?- (A) Find all critical values of f. lfthere are no critical values, enter 1000. lfthere are more than one: enter them separated by commas. Critical value(s) 2 (B) Use interval notation to indicate where f($) is increasing. Note: When using interval notation in WeBWorK, you use | for 00' -| for 00, and U forthe union symbol If there are no values that satisfy the required condition, then enter "{}" without the quotation marks. IncreaSing: (C) Use interval notation to indicate where x) is decreasrng. Decreasrng: (D) Find the xcoordinates of all local maxrma of f If there are no local maXima, enter 4000. If there are more than one: enter them separated by commas. Local maxima at a: : (E) Find the xcoorclinates of all local minima of f. lfthere are no local minima: enter 4000. lfthere are more than one, enter them separated by commas. Local minima at a: = (F) Use interval notation to indicate where x} is concave up Concave up: (G) Use interval notation to indicate where f (@) is concave down. Concave down: (H) Find all inflection points of f. If there are no inflection points, enter -1000. If there are more than one, enter them separated by commas. Inflection point(s) at x = (1) Use all of the preceding information to sketch a graph of f. When you're finished, enter a "1" in the box below. Graph Complete