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modern mathematical statistics with applications
Mathematics And Statistics For Science 1st Edition James Sneyd, Rachel M. Fewster, Duncan McGillivray - Solutions
Muhtar et al (2021) published a Leontief input-output model of integrated farming in Karehkel village in Indonesia. Their model of integrated farming had four compartments; vegetables(mostly lettuce, cai xin and kale), livestock (mostly rabbits and sheep), bokashi (a kind of composted fertiliser
On page 227 we saw an example of a Leontief input-output problem in economics, based on 1947 data from the USA. The answer we got in that example wasn’t all that interesting, but it becomes more interesting when we consider how the economy responds to changes in demand.a. Suppose that the demand
Solve the Leontief input-output problem in the previous chapter(Exercise 10.7) to find the number of pencils, cups of coffee and theorems needed to generate the desired output.
Suppose you have three NaCl solutions, of concentrations 0.1, 0.2 and 0.3µM, and you want to use some of each solution in order to make 0.1 L of a 0.22µM NaCl solution.a. How much of each solution would you use?b. How many ways can you combine your three solutions to get the desired outcome?c.
Solve the mixing problem in Exercise 10.6.
Find the number of each type of flower in Exercise 10.11.
In preparation for the America’s Cup, a dietary supplement is proposed for Emirates Team New Zealand. Altogether they are to have 1920 mg of vitamin D, 310 mg of selenium and 840 mg of folic acid per day. This is to be accomplished by using three liquid food supplements Alpha, Beta and Gamma.
The daughter of one of the authors of this book is getting married in three weeks (as of writing this exercise; true story), and she needs to make up some flower bouquets for her four bridesmaids (also true). She wants to have a total of 200 flowers in each bouquet, and each bouquet needs to
Propane (C3H8) reacts with oxygen (O2) in the air (i.e., it burns)to form carbon dioxide (CO2) and water (H2O). This equation can be written as aC3H8 + bO2 −→ cCO2 + dH2O, for some unknown stoichiometric coefficients,a, b, c andd. By balancing the number of C, H and O atoms on each side of the
Nitric oxide (NO) reacts with oxygen (O2) in the air to form nitrogen dioxide (NO2), which is a major air pollutant. This equation can be written as aNO + bO2 −→ cNO2, for some unknown stoichiometric coefficients,a, b, andc. By balancing the number of N and O atoms on each side of the equation
Suppose we have five web pages in the configuration shown in Fig. 10.7. Follow the method outlined at the beginning of this chapter (page 203) and construct the linking table for this configuration. From this linking table write down the system of equations that can be used to determine which is
Earlier in this chapter (page 208) we had a quick look at Leontief input-output economic models. Here’s another example of such a model.Suppose that we have an "economy" that produces only three items; pencils, coffee and theorems. You want this "economy"to produce 1 pencil, 10 cups of coffee and
Suppose you have two sugar solutions, one of 0.3 M the other of 0.5 M. Set up two equations to determine how much of each solution must be mixed in order to end up with 3 L of 0.45 Msugar solution. Is it possible to get a final solution of 0.6 M?What about a final solution of 0.5 M?
The sum of four numbers is 30. The third number is twice the second, while the first number is one less than three times the fourth number. Write down a set of linear equations that you can use to determine the four numbers. How many solutions do you expect to get?
The sum of three numbers is 30. The third number is twice the second, and is also one less than three times the first. Write down three linear equations that you can use to determine the three numbers.
As we saw in Section 9.4, the two-dimensional rotation matrix is R(θ) =cos(θ) − sin(θ)sin(θ) cos(θ).What is R−1? Check that RR−1 = R−1R = I. What is the geometrical interpretation of R−1, i.e., what does R−1 do to a vector?
Draw a square and describe each of its four corners by a column vector. Multiply each vector by a scaling matrix to double the size of the square, and draw the result. For fun, pick a software package and, using the transformation matrices given above, try rotating, reflecting, or shearing your
The spin is a unique property belonging to quantum particles;the effect of spin in atoms can be observed by deflecting flying atoms using a magnetic field. Spin can be represented by a vector, z, in three dimensions, where zx, zy, zz , are the spin components in the x, y, and z directions. When
Consider the following matrix A =cos π4 sin π4− sin π4 cos π4.Calculate Ax for x = 1 0 and x = 0 1 .a. Sketch the original vectors and the results of the vector product.b. What is the relationship between the original vectors and the vectors you get after multiplying by A? What are you
Given any two vectors, g andc, in two dimensions is it always possible to find a transformation matrix, A, such that Ag = c?Explain your answer carefully. What about if the vectors are in three dimensions?
a. What is the inverse of the matrix A =0 7 9 0?b. Guess the inverse of the matrix B =0 0 7 0 9 0 11 0 0.Check to see whether or not your guess is correct by calculating BB−1 and B−1B.c. How would you generalise this to any matrix with terms only on the
a. What is the inverse of the matrix A =7 0 0 9?b. Guess the inverse of the matrix B =7 0 0 0 9 0 0 0 11.Check to see whether or not your guess is correct by calculating BB−1 and B−1B.c. How would you generalise this to any matrix with terms only on the
a. Let A =5 − λ 0 0 2 − λ.Find all the values of λ for which the determinant of A is zero.b. Let A =5 0 0 2.Find all the values of λ for which the determinant of A−λI is zero (where I is the identity matrix).c. Let A =5 − λ −2 1 2 − λ.Find all the values of λ for which the
Sulphur dioxide (SO2) is a toxic gas and a major air pollutant. The molecule has a planar shape, and is sketched in Fig.8.18, which shows the relative positions of the atoms. The coordinates of each atom are given in units of ångströms, or Å.a. What are the two bond lengths, L1 and L2? Give your
A methylene molecule (Fig. 8.17) contains one atom of carbon(C) and two atoms of hydrogen (H), and is thus written as CH2. The distance from the C atom to each H atom is 1.085ångströms (Å).a. Using the fact that 1 Å is 0.1 nm, write the bond length in metres, using scientific notation.b. If the
As we have seen (Fig. 8.11), the bonds between atoms can be represented by vectors. The atoms in a water molecule (H2O, with atoms O, H1 and H2) are positioned at the coordinates (in units of Å)O = (0, 0, 0), H 1 = (0.950, 0, 0), H 2 = (−0.238, 0.920, 0).a. Use these coordinates to construct the
You’ve probably all experienced the power of recommendation systems. Every time you start up Netflix or Spotify or Amazon or any one of many other online merchants, you are provided with a list of things (movies, songs, books, etc) that the company thinks you might enjoy. The recommendations are
Suppose that a homing pigeon is released directly south of its home, and flies home at a steady speed of 50 km/hour. However, there is a steady wind from the north-east of 15 km/hour. What direction should the pigeon fly in order to make it home by the shortest possible path?
Light absorption in molecules is based on interaction between the electrical oscillation of the light and the electrical dipole oscillation of the molecule. In liquids and solids, molecules are free to move or rotate as they please and the dipole vectors reorient as they do so.Suppose a vector
represent the forces on an interstellar object, O, due to the gravitational effect of three nearby stars. The forces are −→OP,−−→OQ and −−→OR.a. Determine the components of each force in the x and y directions and write −→OP, −−→OQ and −−→OR as coordinate vectors in
An object is said to be in static equilibrium if the sum of all forces applied to it is zero. The vectors in Fig.
Suppose that we know that u · u = 0, for some vector u. What is u?
Suppose that u = (1, 2, −3), v = (3, 0, 2), w = (1, −3, 5) and k = 3. Show thata. u · v = v · u Just like multiplication, the dot product is commutative, i.e., the order doesn’t matter, and is distributive.The dot product is also associative, but only with real numbers, not with other
Let z = (0, −1, −1). Find a vector that isa. parallel to z, with length 5.b. parallel to −z, with length 4.c. orthogonal to −z, with length 3.
Let q = (1, 0, −2). Find a vector, j, that isa. parallel to q, with length 1.1b. parallel to −q, with length 2.c. orthogonal to −q, with length 3.d. orthogonal to q, with length 4.
If u = (1, 0, 2), v = (−1,b, c), calculate b and c so thata. u and v are orthogonal.b. u and v are parallel.c. kvk = kuk.
If u = (1, 2), v = (−2, b), calculate b so thata. u and v are orthogonal.b. u and v are parallel.c. v = −2u.d. kvk = kuk
The discrete Fourier transform (DFT) and the fast Fourier transform (FFT) are possibly the two most important transforms you will ever see. All modern telecommunications relies on the FFT, as does data compression, satellite imaging, medical imaging, and a host of other application areas.
In many models of oscillations in science, the population of one oscillating species (p) can be modelled as p(t) = p0e λt , where λ is a solution of λ2 + βλ + α2 = 0, for some constants α and β, which can represent a wide range of things, depending on the scientific context. They could be
This question is about a simple linear filter. Suppose we have a chemical with concentration C(t). The chemical is being degraded at the rate aC for some constant a > 0, and is being made at the oscillating rate f (t) = a + e iωt. It can then be shown that C(t) = 1 +a + iωe iωt.a. Why did we
An RC circuit is an electrical circuit with a resistor, R, and a capacitor, C, as shown in Fig. 7.12. If the input voltage, V(t),is a given oscillating function of time (t, in units of s), such as V(t) = V0 cos(ωt), the voltage difference across the capacitor (i.e., the output voltage) is given by
Start with z = 1 + i and calculate √z, p√z (i.e., z 14 ), z 18 , etc.What happens as you keep on taking the square root? Plot the sequence in the complex plane. Now repeat the process with z = −1. What happens? Now repeat the process with any complex number. What happens? Does it matter what
The growth of human organs over the first two decades of life sometimes keeps pace with the overall growth of the body, and sometimes doesn’t. For example, Fig. 6.11 shows some data of human body and organ weight over the first 25 years of growth.This is an example of allometry, i.e., the study
Newton’s law of cooling (see Chapter 25) says that, if you put an object with initial temperature T0 K into an environment with a constant temperature A K (where A could be either greater or less than T0), then the temperature of the object can be modelled by T(t) = A + (T0 − A)e−r t, for
Almeida Filho et al (2018) measured how the enzyme chymotrypsin is affected by a compound called LsCTI. (Chymotrypsin is an important digestive enzyme secreted by the pancreas.) They constructed Lineweaver–Burke plots (i.e., lin earised versions of the Michaelis–Menten equation) of chymotrypsin
An experiment was conducted in which a substrate concentration S (in M) was varied and a reaction rate V (in M/s) was measured for each value of S. The following (pretend) data were obtained.The data are expected to follow the equation V(S) = Vm S 3 K3 + S 3 .Using the experimental data, follow the
The rate, V, of an enzyme reaction is a function of the substrate concentration, S, and is described well by the Michaelis–Menten equation,V(S) = Vmax S K + S , for some constants Vmax and K. In practice, we measure V and S and use these measurements to determine Vmax and K.a. If S has units of
Suppose we have a biochemical reaction, where the concentration, s, of substrate is known to be s(t) = ke−At2 eBt, for some constants k, A, and B. Here, t is time (in units of seconds) and s has units of µM.Now suppose you do an experiment to determine the three unknown constants. In this
Cheyne–Stokes respiration is a condition where your breathing has an abnormal pattern, regularly stopping entirely, and then breathing deeply at other times. The cycle repeats with a period of around 0.5–2 minutes. Two different types of Cheyne–Stokes breathing patterns are shown in Fig.
The job of the kidney (you should all have two, but you can get by with just one if you have to) is to filter the blood, removing toxins and waste products, and adjusting the balance of various electrolytes such as Na+, K+, Cl−, H+, and HCO−3. Kidneys also ensure that your body retains the
Most animals (including humans) have a natural circadian rhythm. Although the sun wakes us up in the morning, if the sun didn’t rise we’d still wake up. Our bodies have an internal rhythm that would keep us waking and sleeping with a regular rhythm. However, our natural rhythm is not exactly 24
In Exercise 5.17 we saw how combinations of trig functions could be used to describe oscillations that occurred in layers, with a faster oscillation superimposed on a slower one. In this question we explore another way in which trig functions can be used to model complex oscillatory behaviour.NOAA,
The menstrual rhythm, with a period of close to a month (at least in humans), is another critical physiological rhythm, and female body temperature fluctuates on a monthly cycle. However, these Although the word "menstrual" is derived from the Latin word "mensis"(month), and the Greek word
It has been known for many years that in a predator and prey ecosystem, the numbers of predators and prey fluctuate periodically. For example, red foxes in northern Sweden prey on voles. Studies of these species have shown approximate popThey also eat grouse and hares but appear to prefer voles.
An electrocardiogram (ECG) is a graph of voltage against time of the electrical activity of the heart. It’s measured by placing electrodes on the skin in a number of different places. Changes to the normal electrocardiogram pattern are an important way of diagnosing cardiac problems and diseases.
Electromagnetic radiation (for example, light) has two components, an electric field and a magnetic field, which oscillate as the radiation propagates. The electrical field, E(t), can be described by E(t) = Acos(ωt + φ(t)), where A is the amplitude (with units of V m−1 or N C−1 ), ω is the
Here’s a table of average monthly temperatures at Wellington Airport. Each data point was obtained by averaging over the years from 1971–2000.a. Graph these data. (Time should be in units of months, with t = 0 corresponding to 1 Jan).b. How would you describe these data using a cos function?How
In mammals, circadian rhythms (i.e., the rhythms that occur on the day/night cycle) are controlled by neurons in the suprachiasmatic nucleus of the hypothalamus. Fig. 5.19 shows a result from a model of circadian rhythms showing how the concentration (Vi, in nM) of neurotransmitter produced by a
Plot the following polar functions.a. r = 1 + cos(θ).b. r = 1 + cos(4θ).c. r = 1 + cos(6.4θ).d. r =sin(θ)√| cos(θ) |sin(θ)+7 5− 2 sin(θ) + 2.5.11 The blood pressure P(t) (measured in mmHg) of a typical person is sketched as a function of time in Fig. 5.18.a. Estimate the maximum and
Find all the solutions between −π and π for the following equations, and demonstrate graphically.a. sin(t) = −0.2.b. tan(h) = 0.5.c. cos(2k) = 0.2.d. sin(3d) = −0.3.
Suppose that sin(θ) = 0.5. Without calculating θ, calculatea. cos(θ).b. sin(−θ).c. cos(−θ).d. sin(π − θ).e. sin(θ − π).f. cos(π − θ).g. cos(θ − 2π).h. sin(3π − θ).i. sin(θ − 2π).j. cos(3π − θ).k. cos(θ − 3π).l. sin(2θ).m. cos(2θ).n. sin(3θ − 2π).o.
a. If you want a sector of a circle to have an arc length of 4/5 of the circumference, what must be the angle of this sector?b. If a circle, with radius a mm, has a sector with an arc length of 5 mm, what is the angle of the sector?c. If a circle, with radius b m, has a sector with an area of 5
For each of the following sectors of a circle, calculate its arc length and its area.a. Radius of 1 m, angle of π/2.b. Radius of 1 cm, angle of 3π/2.c. Radius of 2 km, angle of −π/4.d. Radius of 2 km, angle of −3π/4.
What is the amplitude and the maximum and minimum values of the following functions?a. sin(t).b. cos(t).c. sin(5t).d. cos(t/4).e. 3 sin(3t).f. cos(πt)/3.2.g. sin(t) + 17.h. 3 cos(−5t) − 91.i. −7.6 sin(t/2) + π.j. −7.6 sin(At/2) − π.
In response to a flash of light, the voltage across the membrane of a photoreceptor (see the beginning of Section 4.2) will first increase and then decrease. The peak of this flash response tells Typical experimental data can be seen in Figs. 17.1 and 17.12.the brain how bright the flash was.
The Gompertz equation was first derived in 1825 and is now often used to model populations, particularly of tumour cells.One well-known example of this is the work of Anna Laird, who was one of the first to use the Gompertz equation to describe the growth of a population of tumour cells (Laird,
In a random sample of an animal population (containing multiple species) the number of individuals, N, in the sample is related to the number of species, S, in the sample, by the equation S = α ln 1 +Nα, for some constant α, which depends on the species. This equation was first derived by
As we said in the previous question, the Weber–Fechner law applies to all our senses, including smell. An example of this is shown in Fig. 4.12, which is reproduced from Fig. 2 of Wu et al (2016). For a number of odorous compounds they plotted the odour intensity as a function of odour
The Weber–Fechner law says that the relationship between stimulus and perception is logarithmic. Thus, if p denotes perInterestingly, this law seems to apply to all our senses, including taste, touch, sight, sound and smell. ception and S denotes the stimulus, then p(S) = k ln S S0,for some
Let C denote the concentration of paracetamol in your blood.When you take the paracetamol C goes up quickly and then declines slowly over the next few hours, as the drug gets metabolised or excreted.Typical data are shown in Fig. 4.11(Portolés et al, 2003). The"test drug" and "reference drug"
Ion channels sit in the membrane of cells, and allow current to flow from outside the cell to inside, or the other way around.This current is carried by charged ions, such as Na+, K+ or Cl−.Let’s consider a Na+channel, just to be specific (although other ion channels behave the same way). The
Doubling time is an intuitive measure of the rate of growth of a system. For example, it is especially useful for the initial period of growth of a population when growth rate only depends on the number of individuals and not any external pressures such as resource scarcity. Assume that a
The concentration, r, of reactants over the course of a simple chemical reaction is defined by Since r is decreasing along an exponential curve, this is called exponential decay. r(t) = Re−kt, where k is the rate constant, t is time and R is the initial concentration of the reactant. Let k be 100
Suppose that the number of bacteria in a petri dish is given by B(t) = 10,000e 0.1t, where t is measured in hours.a. How many bacteria are present at t = 0, 1, 2, 3 and 4 hours?b. Find the time t, in hours, when the number of bacteria reaches 100,000.
Hydrogen ions are formed when water dissociates into H+and OH−. This process is maintained in a chemical equilibrium at all times, which means that Kw = [H +][OH−] = 1 × 10−14 .If pH is − log10([H +]), then pOH can be defined as − log10([OH−]). Use Kw to calculate the value of pOH for
The normal distribution, G(x), is defined as G(x) =1√2πσ2 e−(x−µ)2 2σ2, where σ > 0 and µ are constants. It’s used in a huge variety of applications to describe measurements of just about anything.a. You can get a rough feel for the function by computing some of the values. Choose any
The Goldman–Hodgkin–Katz equation says that the current density, I (in units of A µm−2), of Na+through a cell membrane can be modelled by the equation I = PN F2V RT "ci − cee−V F RT 1 − e−V F RT #, where F is Faraday’s constant, R is the gas constant, V is the voltage difference
The Swedish physicist/chemist Svante Arrhenius derived the equation which relates the rate of a chemical reaction, r, to the temperature, T, (with units K), the gas constant, R (with units J mol−1 K−1), and the reaction’s activation energy, Ea (with units J mol−1). The Arrhenius equation is
The cross-sectional diameter of a laser as it is focused through a lens is given by d(z) =p az2 + bz + c, where d(z) is the beam diameter (with units of mm), z is the distance (in mm) the measurement is taken relative to the focusing lens, anda, b, c are constant for any particular laser(although
The total surface area, S, (measured in m2) of n metal particles of radius r (measured in m) is given by the equation S = 4πnr2.The number of particles, n, is linked to the mass of one particle, m (in kg) and the total mass of metal, M (also in kg), by the equation M = nm.The mass m (in kg) of one
Suppose that we have three chemical species, A, B and C, where A is converted to B, and B is converted to C. Suppose Remember that square brackets denote a concentration. Thus, for example, [B] is the concentration of B.further that [B] = a[A] +b, and that [C] =[B]2 K2 C+[B]2, where a, b and KC are
Suppose that a particle is sitting at the origin, and, at t = 0 s, starts moving at 3 m s−1 away from the origin for 5 seconds.Then, if we let d denote the distance (in metres) of the particle to the origin, we know that d = 3t, where t is the time in seconds. Now suppose that at time t = 5 s the
Suppose that we are given a function, f , of the temperature, T, defined by f (T) = a + bT + cT2, for some constantsa, b andc, and where T has units of K.a. What is the scientific domain of f ?b. If f is dimensionless (i.e., has no units), what are the units ofa, b and c?
The van der Waals equation isP +an2 V2 V n− b= RT, where P is the pressure of the gas, V is the volume, R is the gas constant, T is the temperature, n is the number of moles, and a and b are parameters that depend on the gas.a. What are the units of a and b?b. Define a new variable, the molar
Suppose that the rate, r, of an enzyme reaction is given by r(c) =Vcc 2K 2c + c 2,where c is the concentration of a chemical (with units of molar, This is called the Hill equation, after Archibald Hill.or mol/L), and Vc and Kc are positive constants. Since r is a rate it has units of mol/s.a. For
For each of the following scientific equations, describe its scientific domain and range, and check that the units agree on both sides of the equation.a. Einstein’s equation that describes the equivalence of mass are.and energy (E = mc2 ).b. The equation for the kinetic energy of a body of mass m
Think of two different functions, g(h) and k(p), such that g(k(m)) = k(g(m)). What about an example when g(h) is not the same as k(h)?
For which of the following functions is b proportional to m?What is the constant of proportionality (if it exists)?a. b = 4m.b. m = 4b − 1.c. b = 4m 2.d. m = −b 2 − 1.e. b = am.f. m = a 2b.g. b = a 3m.h. m =b a+ 4.
At the IAAF World Championships in London, in 2017, the winner of the men’s 200 m sprint was Ramil Guliyev from Turkey, who ran the 200 m in 20.09 s. Guliyev’s 10 m segment times in the final 50 m of the race are given in Table 2.1.a. What was the average speed of Guliyev over the final 50 m of
In this question we compare two ways of handling precision within calculations. The first method involves simply counting the number of significant figures, while the second method uses the rules for calculating the absolute uncertainty. As we shall see, the second method is a lot better than the
The total rate, r (in µmol/s) at which a population of calcium pumps remove calcium from a cell is given by the formula r = dA r¯, where d (in µmol µm−2 ) is the density of the pumps, A (in µm2 ) is the area of the cell membrane, and r¯ (in s) is the time constant of a single pump.Suppose
The concentration of a solution of a strong base like sodium hydroxide (NaOH) can be determined by titration, a technique where a strong acid like hydrochloric acid (HCl) is added slowly until equivalence is reached (i.e., until the amounts of H+and OH−are equal, and thus exactly the right amount
The intensity of light is measured by a photon counting device such as a charge-coupled device (CCD) or photomultiplier tube Errors in measurement using a CCD arise from a number of Noise is a word often used to describe random errors factors. Errors generated by the image itself (due to random
From the ideal gas equation, we know that, for an ideal gas PV = nRT, where P is the pressure of the gas, V is the volume, R is the gas constant, T is the temperature, and n is the number of moles.Suppose that you have 1.01 moles of an ideal gas, at 300.0 K, in a volume of 3.140 0 La. What is the
An experiment to calibrate the volume of a pipette was conducted using water (instead of mercury for safety reasons) and a balance that is precise to 4 decimal places. After 10 weighings, the recorded mass was 4.984 2 ± 0.000 05 g. The density of water on that day was 0.998 9 ± 0.000 05 g
a. Convert the concentration 1 mol m−3 into µmol/Lb. Suppose that a pollutant is present in a completely fictional harbour at a concentration of 0.4µg L−1. Express this in parts per billion (ppb) and parts per million (ppm).c. To obtain the concentration in part (b), a scientist measured
Analytical chemistry uses highly precise glassware, but the volumes indicated may not always be accurate. A common way to improve the uncertainty of an experiment is to calibrate volumes of glassware using a dense liquid and a high precision balance.Two pipettes labelled as 10 mL pipettes dispense
What are the absolute and relative uncertainties in the following calculations?a. P = (−9.34 nm) × (0.49 ms).Make sure you always include the correct units when you calculate the absolute uncertainties.b. v = 5.06 nm 2 ms . What is the value of v in units of m s−1?c. r = 2.00µmol 0.4µL. What
Assuming standard uncertainties, what are the absolute and relative uncertainties in the following calculations?a. 1.13 m + 1.567 m?b. 1.13 m − 1.567 m?c+1.13 g+1.567 g −3.2 g?d+1.13 s −1.567 s +0.43 s?
Calculate the answers to the following calculations, being careful to write the answer with an appropriate number of significant figures.a. 7.3× (2.00 × 10−2)b. (1 × 101) × (1 × 101)c. (1.00×101)× (1.00×103)d. 7.3× 0.02
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