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# f95d28f3-07b7-4b6a-a8e0-571a87f426df - AP CALCULUS...

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AP CALCULUS SUPPLEMENTARY RELATED RATES PROBLEMS : 1) If a snowball melts so that its surface area decreases at a rate of 2 2 min cm , find the rate at which the diameter changes when the snowball has radius 5 cm . 2) Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the pressure P and the volume V satisfy the equation PV C = where C is a constant. Suppose that at a certain instant, the volume is 800 3 cm and is increasing at a rate of 10 3 s cm while the pressure is 100 kPa . At what rate is the pressure changing at this instant? 3) When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV C 1 4 . = where C is a constant. Suppose that at a certain instant, the volume is 500 3 cm and the pressure is 90 kPa and is decreasing at a rate of 5 min kPa . At what rate is the volume changing at this instant? 4) A rocket is launched vertically and is tracked by an observer 3 km away. If the observer’s angle of elevation to the rocket is increasing at 2 o per second, what is the rocket’s velocity at the instant the angle of elevation is π 6 radians? 5) Water is being collected from a block of ice with a square base. The ice is melting in such a way that the edge of the base of the block is decreasing at 4 cm hr , while the block’s height is decreasing at 3 cm hr . At what rate is the water flowing into the pan when the base of the block has edge length 20 cm and the block’s volume is 6000 3 cm ? 6) Water is being poured into a rectangular prism (see diagram to the right) at a rate of 100 3 m hr . What is the rate at which the water level is rising when the height of the water level is 15 m ? 7) Water is being pumped from a square pond with side length 40 m into a circular pond with a radius of 10 m . If the water level in the square pond is going down at a rate of 15 cm min , then how fast is the water level in the circular pond changing? 8) A hemispherical water tank with radius 15 m , depth h meters and volume V cubic meters is defined by the equation ( ) ( ) V h h = π 3 45 2 . If the water tank is full and a plug at the bottom is pulled, then Torricelli’s Law states that dV dt k h = − , where k is a positive constant. Find a formula for dh dt in terms of h only. 9) A point moves along the curve y x 2 3 = in such a way that its distance from the origin increases at the constant rate of 2 units per second. Find dx dt at the point ( ) 2 2 2 , . 10) When two electrical resistors, R 1 ohms ( ) Ω and R 2 ohms ( ) Ω are connected parallel in a circuit, then the effective resistance is 1 1 1 1 2 R R R eff = + . Suppose that R 1 is increasing at a rate of 10 Ω s and R 2 is decreasing at a rate of 2 Ω s . Find the rate at which the effective resistance is changing when R 1 100 = Ω and R 2 400 = Ω .

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f95d28f3-07b7-4b6a-a8e0-571a87f426df - AP CALCULUS...

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