Science One Math. October 17, 2017

Transcription

1 Science One Math October 17, 2017

2 Today A few more examples of related rates problems A general discussion about mathematical modelling A simple growth model

3 Related Rates Problems Problems where two or more variables are related to each other (the relationship is often derived from geometrical arguments) and we want to find the rate of change of one quantity given the rate of change of the other quantity.

4 Problem: As gravel is being poured into a conical pile, its volume V changes with time. As a result, the height h and radius r also change with time. Find how fast the height of the pile is changing when the pile is 2 m high and gravel is poured at 10 m 3 /min.

5 Problem: As gravel is being poured into a conical pile, its volume V changes with time. As a result, the height h and radius r also change with time. Find how fast the height of the pile is changing when the pile is 2 m high and gravel is poured at 10 m 3 /min. Solution: By differentiating the volume formula we get!"!# = & ' 2r!*!# h + r-!.!# we are given!" = 10!# m3 /min, we need!*!#

6 Problem: As gravel is being poured into a conical pile, its volume V changes with time. As a result, the height h and radius r also change with time. Assume the proportions of the pile remain constant as the pile grows and the pile is H m high and D m wide (at the base) when the pouring of gravel stops. Find how fast the height of the pile is changing when the pile is h 1 m high and gravel is poured at k m 3 /min. Solution: By differentiating the volume formula we get!"!# = & ' 2r!*!# h + r-!.!# given!" = 10!# m3 /min, need!*!# assume. -* = 2 3 r = 3-2 h!*!# = 3-2!.!#!"!# = & 4 3 5!. h- 25!# solve for!.!#, evaluate at h = h 1,!"!# = k.

7 Problem: Suppose a V shaped formation of birds forms a symmetric structure in which the distance from the leader to the last birds in the V is r and the distance between those trailing birds is D and the angle formed by the V is θ. Suppose the shape is gradually changing: the trailing birds start to get closer so that their distance apart shrinks at a constant rate k while maintaining the same distance from the leader. Assume that the structure is always in the shape of a V as the other birds adjust their positions to stay aligned in the flock. At what rate is the angle θ changing?

8 Problem: Suppose a V shaped formation of birds forms a symmetric structure in which the distance from the leader to the last birds in the V is r and the distance between those trailing birds is D and the angle formed by the V is θ. Suppose the shape is gradually changing: the trailing birds start to get closer so that their distance apart shrinks at a constant rate k while maintaining the same distance from the leader. Assume that the structure is always in the shape of a V as the other birds adjust their positions to stay aligned in the flock. At what rate is the angle θ changing? Solution: At any instant 3!3 = r sin(θ/2). We are given -!#!3 Ḇ!B = r cos!b = CD!#!#!# * EFG(B/-) = k (constant), r fixed. We don t know θ. We know at any instant cos θ/2 = 1 sin - (θ/2) = 1 D - /(4r - ).!B!# = CD * 1C3 5 /(4* 5 ) = C-D 4* 5 C3 5 As the trailing birds get closer, D r the rate of change of θ k/r

9 Problem: A spider moves horizontally across the ground at a constant rate, b, pulling a thin silk thread with it. One end of the thread is tethered to a vertical wall at height h above ground and does not move. The other end moves with the spider. Find an expression for the rate of elongation of the thread in terms of b and x, the position of the spider at time t.

10 Solution: Let l be the length of the thread when the spider is at a distance x from the wall. know: x and l are functions of time, h is a constant, dx/dt = b, also constant. Need to find: dl/dt when spider is at x. Relationship between x and l: l 2 = x 2 + h 2, true at any instant. Then, by chain rule and implicit differentiation, 2l dl/dt = 2 x dx/dt + 0 dl/dt = (x/l) dx/dt Given dx/dt = b, we get dl/dt = (x/l)b. Finally, l = (x 2 + h 2 ) 1/2 therefore dl/dt = (xb)/ (x 2 + h 2 ) 1/2.

11 Making coffee Coffee is draining from a conical filter into a cylindrical coffeepot at a rate of 10 cm 3 /min. How fast is the level in the pot rising when the coffee is 5 cm deep? How fast is the level in the cone falling then?

12 A sliding ladder A 5-m ladder is leaning against a house when its base starts to slide away. By the time the base is 4 m from the house the base is moving at the rate of 2 m/s. How fast is the top of the ladder sliding down the wall then? At what rate is the area of the triangle formed by the ladder, wall, and ground changing then? At what rate is the angle θ between the ladder and the ground changing then?

13 Walkers A and B are walking on straight streets that meet at right angles. A approaches the intersection at a speed a m/s; B moves away from the intersection at b m/s. At what rate is the distance between the walkers changing when A is 10 m from the intersection and B is 20 m from the intersection?

14 An airplane problem An airplane climbing at an angle of 45 o passes over a ground radar station at an altitude of 8 km. At a later time the distance from the radar station to the airplane is 9 km and is increasing at the rate 700 km/h. Find the speed of the airplane at that time. (Hint: draw a perpendicular from the radar station to the flight path of the plane)

15 Summary: Solution strategy for related rates problems Draw a diagram List time-dependent variables and constants List given data Identify what you need to find Write down a relationship between the variables whose rate you need to find and that whose rate is given Use implicit differentiation and chain rule If required, evaluate rate of interest at specific instant.

16 Mathematical Modelling A mathematical model is an attempt to describe a natural phenomenon quantitatively. No model is an exact representation of the phenomenon: models of natural systems always incorporate assumptions regarding the system being modeled. E.g: A model for free fall:!k = g assumptions: no air resistance,!# constant gravity When assumptions are not met, the model may be a poor representation of reality model must be refined.

17 The entire spectrum of mathematical models is broad. Basic distinction: algebraic versus evolution equations. E.g. Growing cube crystal V = x ' (volume of a cube of side length x) algebraic equation governing what would happen if a quantity is changed (if x is twice as long, V is 8 times larger, etc.) Doesn t say how the quantities change.!"!# = 3x-!P!Q model equation describing how quantities of interest change over time, evolution equation Evolution equations are derived from basic principles

18 A simple model for uncontrolled growth P(t) = # of individuals in population at time t, b = per capita birth rate (per unit time) m = per capita mortality rate (per unit time). Assumptions: b and m are constant over time. Change in population over a time interval Δt is P: P = P t + t P t = [# of births] [# of deaths] balance equation P t + t P t = b P t m P t Δtà0,!W!# = b m P W #X # CW # # = (b m)p Let r = b m, net per capita growth rate (per unit time): (intrinsic parameter) dp dt = rp

19 Malthusian Model (or exponential growth) Uncontrolled growth model: dp dt = rp Solution (guess and check): P t = P(0)e *# Simplistic model, reasonable for population with unlimited resources (food, land, etc.) (Thomas Malthus, An Essay on the Principle of Population, 1789)

20 Exponential Growth Models P t = P(0)e *# for some constant r P 0 provided P 0 0 if r > 0 growth if r < 0 decay Two main properties: 1. Fixed relative growth rate 2. Fixed doubling time (or half life)

21 Science One Math October 18, 2017

22 Today Other examples of exponential growth and decay models

23 Warm up: which of the following equations has solution y t = 3e -#? A. B. C. D.!Q!# = 3y!Q!# = 3e-# To check whether a function y is a solution!q!#!q!# = 3t to a differential equation, compute y and = 2y plug into equation.

24 A population of animals has a per-capita birth rate of b = 0.08 per year and a per-capita death rate of m = 0.01 per year. Assuming b and m are constant, the population size P t is found to satisfy which one of the following equations? A.!Q!# = 7y!Q B. = 0.07y!#!Q C. = 0.09y!# D. P t = P 0 e i# E. P t = P 0 e k.ki#

25 A population of animals has a per-capita birth rate of b = 0.08 per year and a per-capita death rate of m = 0.01 per year. Assuming b and m are constant, the population size P t is found to satisfy which one of the following equations? A.!Q!# = 7y!Q B. = 0.07y!#!Q C. = 0.09y!# D. P t = P 0 e i# E. P t = P 0 e k.ki#

26 Exponential Growth Models!W!# = rp has solution P t = P(0)e*# for some constant r P 0 provided P 0 0 if r > 0 growth (birth rate > mortality rate) if r < 0 decay (birth rate < mortality rate) Two main properties: 1. Fixed relative growth rate 2. Fixed doubling time (or half life)

27 Property 1: Fixed relative growth rate dp dt is the growth rate (per unit time) changes over time depends on the size of population mn mo W = r = % growth rate (per unit time) (relative to the size of population) constant over time

28 Property 2: Fixed Doubling Time Suppose P t = P(0)e *# given some initial condition P(0) = P k. Let τ = time it takes P to double in value (doubling time) Does τ depend either on time or population size? P τ = 2P 0 = 2P k P 0 e *q = P k e *q = 2P k e *q = 2 rτ = ln 2 (Doubling time) τ = ln 2 r constant for a specific population

29 Example: World population Data: 2 billion in 1927, 4 billion in 1974 Problem: Make a prediction for the world population in 1999 and Solution: Assumption: exponential growth!w t in years since 1927, P in billions. 1927: t = 0, P 0 = 2!# = rp, r is constant over time, 1974: t = = 47,P 47 = 4 doubled! r = xy - 4i = ~1.5%, our model is P t = P(0)e[(~-)/4i]# Predictions: 1999, t = = 72, P 72 = 2e 5 ƒ 5 ~ 5.78 ƒ , t = = 89, P 89 = 2e ~ 7.43

30 Bacteria population Assume that the number of bacteria at time t (in minutes) grows exponentially. Suppose the count in the bacteria culture was 300 cells after 15 minutes and 1800 after 40 minutes. What was the initial size of the culture? When will the culture contain 3000 bacteria?

31 Modelling rates of decay: Radioactivity N t = # of atoms in sample at time t k = probability of decay (per unit time) constant for a given chem. element Change in amount of material over a time interval Δt N = N t + t N t = k N t #X # C # # = k N t 0 dn dt = kn where k is a constant

32 In 1986 the Chernobyl nuclear power plant exploded, and scattered radioactive material over Europe. Of particular note were the two radioactive elements: iodine-131 (half-life 8 days) and cesium-137 (half-life 30 years). Which of the following differential equations can be used to predict how much of iodine-131 would remain over time? Let y(t) be the amount of iodine-131 at time t. A. y = y B. y = y C. y = 8.00 y D. y = y E. y = 30.0 y

33 In 1986 the Chernobyl nuclear power plant exploded, and scattered radioactive material over Europe. Of particular note were the two radioactive elements: iodine-131 (half-life 8 days) and cesium-137 (half-life 30 years). Which of the following differential equations can be used to predict how much of iodine-131 would remain over time? Let y(t) be the amount of iodine-131 at time t. A. y = y B. y = y C. y = 8.00 y D. y = y E. y = 30.0 y

34 In 1986 the Chernobyl nuclear power plant exploded, and scattered radioactive material over Europe. Of particular note were the two radioactive elements iodine-131 (half-life 8 days) and cesium-137 (half-life 30 years). A solution of the differential equation y = k y where k = is y = y 0 e t iodine-131 at time t What are the units of the constant k = ? A. days B. 1/days C. years D. 1/years E. k is a dimensionless number

35 In 1986 the Chernobyl nuclear power plant exploded, and scattered radioactive material over Europe. Of particular note were the two radioactive elements iodine-131 (half-life 8 days) and cesium-137 (half-life 30 years). A solution of the differential equation y = k y where k = is y = y 0 e t iodine-131 at time t What are the units of the constant k = ? A. days B. 1/days C. years D. 1/years E. k is a dimensionless number

36 How long it would take for iodine-131 to decay to 0.1% of its initial level? y=y 0 e t (where t is measured in days) A. About 80 days B. About 53 days C. About 27 days D. It depends on the initial level of iodine-131

37 How long it would take for iodine-131 to decay to 0.1% of its initial level? y=y 0 e t (where t is measured in days) A. About 80 days B. About 53 days C. About 27 days D. It depends on the initial level of iodine-131

38 Suppose that 1000 rabbits are introduced onto an island where they have no predators. During the next five years the rabbit population grows exponentially. After the first 2 years that population grew to 3500 rabbits. After the first 5 years a rabbit virus is sprayed on the island and after that the rabbit population decays exponentially. 2 years after the virus was introduced, the population dropped to 3000 rabbits. How many rabbits will there be on the island 10 years after they were introduced?

39 Suppose an initial dose of 100 mg of a drug is administered to a patient. Assume it takes 36 hr for the body to eliminate half of the initial dose from the blood stream. Find an exponential function that models the amount of drug in the blood at t hr. (your answer should have no unknown constants) At what rate (in mg per hr) is the drug eliminated from the blood stream after 36 hr since the initial dose? If a second 100-mg dose is given 36 hr after the first dose, how much time is required for the drug-level to reach 1mg?

40 Log-linear models P(t) = P(0) e rt exponential models can be converted to a linear model using logs: ln P = ln (P(0) e rt ) ln P = ln (P(0)) + ln( e rt ) ln P = ln (P(0)) + r t y = A + r t