Question 1 A parcel of air rising quickly in the atmosphere will decrease in temperature and increase in volume if it does not exchange heat with the surrounding air. For sufficiently dry air, the relationship between temperature and volume is given by TV"-4 = C for a constant C, temperature ? in Kelvin, and volume V in cubic meters. Let time t be in hours. (a) Find and explain what it represents. Be sure to include units. (b) Find - and explain what it represents. Be sure to include units. (c) Find #1 assuming that both ? and V are functions of time, and explain what it represents. Be dt sure to include units. dt (d) Find- if V = 10 m3, T = 295 K and volume is increasing at a rate of 1 m' every hour. (e) Find ak dt - assuming that both T and V are functions of time, and explain what it represents. Be sure to include units. (f) Find - if V = 10 m, T = 295 K and temperature is increasing at a rate of 2 K every hour. Question 2 According to the ideal gas law pressure, P (in pascals) , volume, V (in cubic meters), and temperature, T (in kelvina) are related by the equation PV = ART where R is the ideal gas constant and a is the number of moles of the gas present. (a) Find assuming that both volume and temperature are changing in time. (b) Create a model for temperature assuming that it varies sinusoidally in time, starting with a minimum temperature of 300 K at * =0 hours, and at its maximum of 320 K at t = 12 hours. (c) Create a model for volume assuming that starts at 10 cubic meters at t = 0 hours and increases by 10% every six hours. (d) Find dt -18 under the assumptions of parts (a), (b) and (c). Your answer will contain a and R. Question 3 The relative rate of change of a function f is given by the ratio . The relative rate of change puts the additive rate of change into perspective compared to the current value of the function. After all, adding 100 people per year is much more significant for a town of hundreds than for a city of millions. (a) Compute the relative rate of change for a power function, that is, a function of the form / (x) = Ar" for positive constants A and n. Analyze the relative rate of change as > > co, and interpret the result. (b) Compute the relative rate of change for an exponential function of the form f(x) = Acks for positive constanta A and k. Analyze the relative rate of change as I > co, and interpret the result. Question 4 A traffic patrol helicopter is stationary a quarter of a mile directly above a highway. Its radar detects a car whose line-of-sight distance from the helicopter is half a mile and is increasing at the rate of 57 mph. Is the car exceeding the highway's speed limit of 60 mph? Justify your answer completely. Question 5 Consider an electrical circuit. The voltage V (volts), current / (amperes), and resistance R (ohms) are related by Ohm's law, which states V = / R. Suppose that V is increasing at the rate of 2 volt/sec while I is decreasing at the rate of = amp/sec. (a) Find the equation that relates un dh to de and dt dt (b) Find the rate at which R is changing when V = 12 volta and / =2 amps. Be sure to include units. la R increasing or decreasing