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# assignment3sol - ma2176 a3 sol Applications of Derivatives...

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1 ma2176 a3 sol Applications of Derivatives 1. An airplane, flying horizontally at an altitude of 1 km, passes directly over an observer. If the constant speed of the plane is 240 km per hour, how fast is its distance from the observer increasing 30 seconds later? Solution : h km 1km s km The relation between h and s is 2 2 1 h s + = . Differentiate both sides of 2 2 1 h s + = with respect to t , we have dt dh s h dt ds dt dh h dt ds s = = 2 2 . Since km 2 120 1 240 km/hr, 240 = = = h dt dh and 5 5 2 1 2 2 = = + = s s . Thus km/hr 5 480 = dt ds . 2. Water is pumped at a uniform rate of 2 liters per minute into a tank shaped like a frustum of a right circular cone. The tank has altitude 80 centimeters, and its lower and upper radii are 20 and 40 centimeters respectively. How fast is the water level rising when the depth of the water is 30 centimeters? (Hint: The volume V of a frustum with altitude l and lower and upper radii a and b is () 22 1 3 Vl a a b b π =+ + ) Solution : frustum water h 20 2 x (removed part) Suppose the depth of water is h and the radius of the surface of water is d. Using properties of similar triangles, we have x x h d + = 20 . In addition, we know that the tank has altitude 80 centimeters and lower and upper radii of 20 and 40 centimeters respectively. Using properties of similar triangles again we have

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2 80 1 80 2 80 20 40 = + = + = x x x x Thus, we have 20 4 80 80 20 + = + = h d h d . The relation between the volume of water V and the depth of water h is : () 22 1 3 Vh d d b b π =+ + , where d is the radius of the surface of water and b is lower radius of the frustum. . Since 20 , 20 4 = + = b h d , we have 2 1 20 20 20 400 34 4 hh  + + +   . Now differentiating V with respect to t gives 2 11 1 1 20 20 20 400 2 20 20 . 4 3 4 4 4 dV dh h h h dh dh h dt dt dt dt ππ  + + + + + +   Put 2000 and 30 dV h dt == into the above identity, we have 2 13 0 3 0 1 3 0 1 1 2000 20 20 20 400 30 2 20 20 4 3 4 4 4 1 2000 756.25 550 400 10 13.75 5 568.75 187.5 3 2000 cm min 756.25 dh dh dh dt dt dt dh dh dh dh dh dt dt dt dt dt dh dt + + + + + + ⇒= + + + + = + 3. A particle P is moving along the graph of 2 , 4 2 = x x y , so that the x- coordinate of P is increasing at the rate of 5 units per second. How fast is the
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assignment3sol - ma2176 a3 sol Applications of Derivatives...

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