B. 1. Starting from the same point, Reden starts walking eastward at - ft/s while Neil starts 2 running towards the south at 6 ft/s. How fast is the distance between Reden and Neil increasing after 2 s? 2. A woman standing on a cliff is watching a motor boat through a telescope as the boat approaches the shoreline directly below her. If the telescope is 250 feet above the water level and if the boat is approaching the cliff at 20 ft/s, at what rate is the acute angle made by the telescope with the vertical changing when the boat is 250 feet from the shore? 3. A right circular cylindrical balloon is being inflated in such a way that the radius and height are both increasing at the rate of 1 in/s and 3 in/s, respectively. What is the rate of change of its total surface area when its radius and height are 20 in and 50 in, respectively? 4. A right circular cone with radius 3 cm and height 6 cm is submerged point first into a tall cylinder of radius 5 cm that is partially filled with water. How fast is the water level rising when the cylinder is completely submerged? 5. A baseball diamond has the shape of a square with sides 90 ft long. A player 60 ft from the second base is running towards the third plate at a speed of 28 ft/min. At what rate is the player's distance from the home plate changing?=% [ Find how fast the volume of a sphere increases as the radius increases. . If water is being drained from a swimming pool and V' liters is the volume of the water at the time minutes after the draining starts, where V' = 250(1600 80 + 2), how fast is the water fowing out of the water pool 5 minutes after the draining starts? . Find the slope of the tangent line at each point of the graph of y = ! + ' 3r* where the rate of change of the slope is equal to zero. . An airplane is flying on a horizontal path at a height of 3800 ft. At what rate is the angle of elevation # of the airplane from P changing with respect to the distance between the airplane and a fixed point P on the ground if 8 = 307 . If two resistors with resistance R and H> are connected in parallel, the total resistance 1 1 R in ohms is given by R-R - E If By and R are increasing at 0.4 ohms/s and 0.25 1 ohms /s, respectively, how fast is R changing when R, = 600 ohms and R; = 400 chms? DATE : ./20 MT WTF S rate of change of volume as radius changes Differentiating V w. v. to "v gives dv . d ( 4 1 X 3 ) dy du = 411. 3 7 3 - 1 dv dy The dul = 41 1 2 tells us that the rate at which volume l of sphere increases wirth respect to -radius is chirectly proportioned to the square of the radius with constant of Proportionality Thus for a given radius "r." The volume increased at a rate of Trz cubic Per unit increase in the radius . CS Scanned with CamScannerB Palt. (1 ) To find how fast the volume of a sphole increases as The radius increases we need to determined the rate of change of the volume with respect to the radius This is a problem that involves calculus specifically the use of derivatos to find the rates of change. the volume of sphone is given below y V = where rIS The radius of the sphere , It is constant approximately equal to 3. 14 159 10 find how fast the volume Increases writ to radius we need to complete the derivative of volume wy.y to radius . The derivative is aver and represents instantonions CS Scanned with CamScannerDATE : / 20 NOWDOS So rate of change of slope 15 0' when K- -1 and kal Now find slope . at x = - 1 dy 4 x3 + 3x2 - 6x = 4(-1)- + 3(-1 ) _ 60-1) = - 4 +3 +6 - 4 +9 : 5 (So slope at x =- 1 15 5 Now find slope at x - dy 4 ( 1 83 + 3 ( 4 ) - 16( 2 ) 2 4 + 3 / 4 - 3 2 +3 - 3.4 X So slope cit war is - 1 2 CS Scanned with CamScannerS lets find derivative of gunon function y= x 4 + x3- 3 12 wr.# %." 4 = 1 4 x x 3 - 3 x 2 ly , - d ( 1 4 + 13 3 x 2 ) 47 3 + 3 0 1 2 _ 6 x Now find again derivative of (is writx . d do \\- d (4 23 + 3x 2 - - 6 x ) on dry 12 x + 6x - 6 du 2 Now set d'yl , , =0 12x + 6x - 6 = 0 . . RX - X = X 2 x 2 + X - 1 - 0 = 2x2 + 2X-X -150 2 x ( 1 + 1 ) + 1 ( x2 + 1 ) 20 (4 + 1 ) 1 2 X - 1 1 -0 x = - 1 x=L 2 CS Scanned with CamScannerDATE : 1 20 -0 . 25 8 /5 into the expression for dR : olt 2402 2410 2 60 02 X 0.4+ 40 02 X 0.25 57602 X 0.4 + 5 7600 x0.25 3 60000 160400 - 0.064 4 0.09 2 0.154 ohms/s So 0. 15 4 onmy CS Scanned with CamScannerUALL : 20 MTWTF S 5 To determine rate of change of the total Resistence d+ R in parallel circuit composed of Resistor R, and R2 we start with formula: K - R + RI w.r.t Differentiating both way by +'time - 1 QR dRI - 1 dRez solving for OR / of we gel dRI R olt RI = boom R 2 = 40 012 60o you = 240 0hm Now Inserting RI = boost. R. = 40012 dRI = D. 4 9 1 CS Scanned with CamScanner