Math 140 Written Homework 7 3 7 3 8 3 9 3 10 Page 1 of 2 From the questions below please choose and solve any 5 problems only The homework is worth 10 points Each question is worth 2 points Show all of your work and put a box around your nal answer Number each attempted question clearly Write legibly that is suitably large and suitably dark if the grader can t read your answer it s consider uncompleted Question 1 Find the derivative of each function a f x ln cid 19 cid 18 x 1 x3 1 b g x x x2 1 x 1 c h x xx2 Question 2 The function M t a b a ekmt 1 m is known as the von Bertalan y function and was introduced in the 1930s by Austrian biologist Karl Ludwig von Bertalan y Calculate M cid 48 0 in terms of the constants a b k and m Question 3 The energy measured in ergs associated with an earthquake of moment magnitude Mw are related by log10 E 16 1 1 5Mw Calculate dE dMw for M 2 5 8 Question 4 A particle moves counterclockwise around the ellipse with equation 9x2 16y2 25 a In which of the four quadrants is dx dt 0 Explain b Find a relation between dx dt and dy dt c At what rate is the x coordinate changing when the particle passes the point 1 1 if its y coordinate is increasing at a rate of 6 m s Question 5 Let L f h be the amount of water in liters a species of tree needs per week when it is h centimeters tall Some values are given in the table below h f h 100 20 150 30 200 42 250 58 300 75 Let h g t 100 0 1t2 be the height in centimeters of a particular tree of this species t months after January 1 2020 Assume this model is valid for the rst 6 years of the tree s life a Estimate f cid 48 250 Give units Explain the practical meaning in a sentence b Calculate g cid 48 39 Give units Explain the practical meaning in a sentence c Estimate Give units Explain the practical meaning in a sentence cid 18 d dt f cid 0 g t cid 1 cid 19 cid 12 cid 12 cid 12 t 39 Question 6 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 su ciently dry air the relationship between temperature and volume is given by T V 0 4 C for a constant C temperature T 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 dT dV dV dT dT dt c Find assuming that both T and V are functions of time and explain what it represents Be sure to include units dT dt dV dt dV dt dP dt Math 140 Written Homework 7 3 7 3 8 3 9 3 10 Page 2 of 2 d Find if V 10 m3 T 295 K and volume is increasing at a rate of 1m3 every hour e Find 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 m3 T 295 K and temperature is increasing at a rate of 2 K every hour Question 7 According to the ideal gas law pressure P in pascals volume V in cubic meters and temperature T in kelvins are related by the equation where R is the ideal gas constant and n is the number of moles of the gas present P V nRT 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 the initial time of t 0 hours and is 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 d Find under the assumptions of parts a b and c Your answer will contain n and by 10 every six hours dP dt cid 12 cid 12 cid 12 cid 12 t 18 R Question 8 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 signi cant for a town of hundreds than for a city of millions f cid 48 f a Compute the relative rate of change for a power function that is a function of the form f x Axn for positive constants A and n Analyze the relative rate of change as x and interpret the result b Compute the relative rate of change for an exponential function of the form f x Aekx for positive constants A and k Analyze the relative rate of change as x and interpret the result Question 9 The volume of a sphere of radius r is V 4 3 r3 If the radius is expanding at a rate of 14 inches per minute at what rate is the volume changing when r 8 in Question 10 Sonya and Isaac are in motorboats located at the center of a lake At time t 0 Sonya begins traveling south at a speed of 32 mph At the same time Isaac takes o heading east at a speed of 27 mph a How far have Sonya and Issac traveled after 12 min b At what rate is the distance between Sony and Issac changing after 12 min Question 11 A tra c 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 12 Consider an electrical circuit The voltage V volts current I amperes and resistance R ohms are related by Ohm s law which states V IR 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 to and dV dt dI dt 1 4 dR dt b Find the rate at which R is changing when V 12 volts and I 2 amps Be sure to include units Is R increasing or decreasing