Question 1 (Direct and Inverse Proportionality) When a chemical solution is diluted, the total number of moles present in the material of the solution remains the same. This relationship can be expressed as C1V1 = C2V2 where C1 is the initial concentration, V1 is the initial volume, C2 is the final concentration after dilution and V2 is the final volume. (a) You are given a solution with a known volume and concentration. (i) Rearrange the equation to find an appropriate expression for the final concentration after dilution. (ii) Is the final concentration directly or inversely proportional to the final volume? (Be sure to fully justify your answer.) (b) 20 ml of a glycine solution with concentration 0.2 mol L−1 (moles per litre) is to be made up to a final volume of 1.6 litres with water. Find the concentration of glycine in the final solution, and express your answer in scientific notation. [3+3 = 6 marks] Question 2 (Linear Functions) Australia adopted the Celsius scale under the Metric Act of June 12, 1970. In the Celsius scale, water freezes at 0◦C and boils at 100◦C (at standard atmospheric pressure). Prior to the Metric Act, the Fahrenheit scale was used for measuring temperature. In this scale water freezes at 32◦F and boils at 212◦F (at standard atmospheric pressure). Although we can just look it up on the internet, we are going to derive a linear function C = g(F ) that converts a temperature F degrees Fahrenheit to C degrees Celsius. (a) Write down the ordered pairs (F,C) corresponding to the freezing and boiling points of water. (b) Find the slope of the linear function. (c) Use the ordered pair corresponding to the freezing point of water and the point-slope form of a linear function to derive the conversion formula. (d) Find and interpret the vertical intercept of the linear function you found in (c). (e) It took until the end of the 1970s before all weather forecasts in Australia were given only in Celsius degrees. Many older people still talk about hot days exceeding the ‘century mark’. Use your formula to determine what 100◦F is in degrees Celsius. (f) Is there any temperature which has the same numerical value in both Celsius and Fahren- heit? Find it algebraically. (g) Sketch a graph of the linear function, showing all important features. [1+3+2+2+1+3+3 = 15 marks] Question 3 (Quadratic Functions) The simplest chemical reaction is when two substances combine to form a third. This is written k as A + B −→ X. In an autocatalytic reaction, the resulting product is used in the formation of the new product, as in the reaction k A + X −→ X. If we assume that this reaction occurs in a closed vessel then the reaction rate is given by R(x) = kx(a − x) where a is the initial concentration of substance A, and k is a positive constant. The variable x is the concentration of the new product X during the reaction. (a) Find the coordinates of the vertex of R(x). Is it a maximum or a minimum? What significance does it have in the context of the problem? (b) Find the x intercepts. Interpret these within the context of the problem. (c) Given that the reaction rate R(x) must always be positive, write down the domain of the function. (d) Sketch the graph showing all important features. Question 4 (Translations and reflections) [6+5+2+2 = 15 marks] Thegraphbelowshowsf(x)=√x. Thedomainoff(x)isx≥0. Therangeoff(x)isy≥0. The graph also shows g(x), which has been obtained through translations and reflections applied to f(x). ￼￼￼-3 -1 y ￼4 ￼3 ￼￼2 1 2 3 4 f(x) ￼￼4 -2 1 ￼￼￼￼￼￼￼￼￼￼￼-1 ￼-3 g(x ) x ￼-2 ￼￼-4 ￼￼￼￼￼￼￼√ (a) Describe, in words, the translation and reflection of f(x) = to obtain g(x). Use your description to write a formula for the function g(x). State the domain and range of g(x). (b) A new function, h(x), is to be a vertical translation of f(x), shifting it one unit down the vertical axis. Write a formula for h(x). State the domain and range of h(x). √ ￼x that have been applied ￼(c) Sketch the graph of p(x) = 3 − (i) Find the x-intercept. (ii) Find the y-intercept. x + 1 by using the following steps. (iii) Use translation and reflection properties to sketch a graph of p(x). It does not need to be to scale. Be sure to label all important points and features. [6+3+6 = 15 marks] Question 5 (Exponential functions) ￼One of the most famous antibiotic drugs is penicillin — the first ‘miracle drug’. These antibiotics were among the first drugs to be effective against many previously serious bacterial infectious diseases, such as syphilis and those caused by staphylococci and streptococci. Adelaide can take some small credit in the discovery of penicillin; Howard Florey (born and educated in Adelaide) shared the Nobel Prize in Physiology or Medicine in 1945 with Chain and Fleming for its discovery. In an average individual, a dose of penicillin will be broken down in the blood so that one hour after injection only 60% will remain active. Suppose a patient is given an injection of 300 milligrams at 12 pm. (a) Consider the following table for the amount of penicillin that will remain in the blood at hour intervals from noon until 5pm. Copy this table into your assignment. Explain the calculation corresponding to t = 2 hours. Complete the remaining calculations. Give answers to one decimal place. Time since injection (t hours) 0 1 2 3 4 5 Amount of penicillin (p mg) 300 0.6×300=180 0.6×(0.6×300) = 300×(0.62) = 108 ￼￼(b) Produce a scatterplot of the data in EXCEL. (c) By observing the pattern in your table, write a function in the form p(t) = a(bt) that can be used to calculate the amount of penicillin p (in milligrams) remaining in the bloodstream after any number of hours t since the injection. (d) Medical scientists are usually interested in the time it takes for a drug to be reduced to one half of the original dose. This is called the half-life of the drug. Mark the half-life of penicillin on your chart (by hand is fine). Give the numerical value of the half-life to the nearest 30 minutes. (You might need to adjust the horizontal scale.) One advantage of the formula in (c) is that we can easily identify the hourly rate of elimination of the drug. However, bases other than the common ones (e, 10 and 2) are not easy to work with. We will write a new equation to model the elimination of penicillin involving the number e. This equation is in the form p = 300(ekt), where k is a constant indicating the continuous rate of elimination. (e) Use the equation p(t) = 300(ekt) to verify that 300 mg of penicillin is administered in the initial injection. (f) By using one data point from your table in (a), find the value of k to two decimal places. (g) Graph your function using EXCEL over 10 hours. Begin at t = 0. Use a line graph. Choose an appropriate step size in order to get a smooth curve. (h) Use p(t) = 300(ekt) and the value of k obtained in (b) to find the half-life of penicillin to two decimal places. (You might also like to fit an exponential trend line to your scatterplot in (a) to check your answer—this is not required.) (i) Use your equation to find the time when less than 10mg of penicillin remains in the bloodstream.
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