1. Ideal Gas Law: PI" = k?" where P is pressure, V is volume, T is temperature (in kelvins) and it is a constant of proportionality. A tank contains 2000 cubic inches of nitrogen at a pressure of 26 pound per square inch and a temperature of 300 K. a. What is k for this tank? b. Write P as a function of V and T. c. Find the following partial derivatives and evaluate them at the given values. Carefully explain what they are telling you. {use correct units) 6P _ E (\"W and T=200 5T T2200 I':IDO i-'=1{]D 2. The area of a circular sector is given by the formula A = 1730 2 where r is the radius and O is measured in radians. H a. For a circle of radius 75 inches, find the area of the sector whose central angle is W /N b. Using a differential, estimate the change in area if the radius decreases by .5 inches and the angle increases by . 1 radians. (Give the answer with the correct units) c. What would the actual change in Area be? d. How fast is the area changing if the radius is increasing at a rate of 1 inch/sec and the angle is decreasing at a rate of .2 radians/sec? (Give the answer with the correct units)The Doppler effect [or the Doppler shift} is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. A common example is the change of pitch heard when a vehicle sounding a horn approaches and recedes from an observer. Compared to the emitted frequency, the received frequency is higher during the approach, identical at the instant of passing by, and lower during the recession. The reason for the Doppler effect is that when the source ofthe waves is moving towards the observer, each successive wave crest is emitted from a position closer to the observer than the crest of the previous wave. Therefore, each wave Lawn- LII"! II: hFr takes slightly less time to reach the observer than the ~ ': i I "my previous wave. Hence, the time between the arrivals of f'H'd successive wave crests at the observer is reduced, + ' .'. causing an increase in the frequency. Conversely, if the i 130\" source of waves is moving away from the observer, each wave is emitted from a position farther from the observer than the previous wave, so the arrival time between successive waves is increased, reducing the frequency. 3. If f; is the frequency of sound produced by a source traveling with speed v3 and an observer is traveling with speed v0 along the same line from the opposite direction toward the source, then the frequency of the sound j; heard by the observer is modeled by the formula: C + V0 .5 = speed ofsound at 0 C = 332 mfs : T4166 mfl'ftr fezf; C VS Suppose at a particular moment, you are in a train traveling at 34 m/s and accelerating at 1.2 mfsf. A train is approaching you from the opposite direction on the other track at 40 m/s, accelerating at 1.4 m/sz, and sounds its whistle, which has a frequency of 460 Hz. At that instant, what is the perceived frequency j; that you hear and how fast is it changing