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1.Â&nbspIn your chemistry courses, you will encounter the term â

    1. In your chemistry courses, you will encounter the term ‘pH’, whichmeasures acidity of a solution.  Neutralis 7, above 7 is labeled Basic, and below 7 is Acidic. We define:  pH = – log [H+](which we read ‘Concentration of hydrogen ions in moles per liter of solution’).A mole is approximately 24! ions, though you don’t need to know that for thisproblem.  But you do need to know thesefacts:  I: The pH of human blood is 7.4;  II: The pH of Coca Cola is 2.5; III: Distilledwater has pH 7;  IV: When we write ‘log’we mean base-10. IV: One ml (milliliter) is 10-3 liters.(i) Thehuman body has a blood volume of 5 liters. So how many moles of H+ are in your bloodstream?  Write the answer in scientific notation.(First figure out how many moles per liter!)(ii) Nowsuppose we make a vampire cocktail punch by mixing blood of 199 humans with 5liters of Coke. Find (a) the total number of moles of H+, and (b)the total volume of punch being made for the party.  Then (c) compute the pH of the punch.(iii) Next,let’s deal just with Coke and water. We want to make a very flat punch for aparty attended by accountants. If we mix xliters of water with 1 ml of Coke, what is the volume of punch, and how manymoles of H+ ion does the punch contain? (iv) Use(iii) to write a formula for the pH of the punch (in terms of x).(v) Nowuse your work in (iii) to determine how many liters of water you will need toadd to 1 ml of Coke, to increase the pH of the punch by one unit above that ofstraight Coke.(vi) Expressthe answer to (v) in a complete declarative English sentence.2.   Suppose we have a population that increasesaccording to the formula P(t) = P0ekt.  Our goalis to fill out the following table:PointtP(t)A0B1020C3050D40E2000(i) Usepoints A and B to write two equations. Then use them to solve for P0 and k.  Finally, write thepopulation equation.(ii) Use(i) to fill in the rest of the chart. You may round each value to the nearest tenth.(iii) Nowuse your formula to determine the doubling time of this population.(iv) Explainhow, without using an equation, you can now calculate the t values for which P(t) equals1000. And how about 250?  Do this bywriting a clear brief essay, with complete English sentences.(v) Nowon graph paper, draw a clear graph of this exponential function. Be sure to labelyour axes, and to label, with their coordinate, each of points A through E.3.  (i) Solve exactly:  22x-1= 11x+1.  (Hint: It isprobably easiest to start by taking logs of each side.)(ii) (a) Consider the function f(x)= ln (x – 1). Find its domain.  (b) Now find the composition f ( f(x) ). Again, figure out itsdomain. (It may help to draw one of those flow diagrams we use in class forcompositions. Work backwards.  (c) Once more. Extend yourwork to find the domain of f(f(f(x))).4.  Consider the followingsituation:  A circle of with center O(0,0), radius 10m, is inscribed in asquare. The ray of angle 30O, in standard position, intersects thecircle at point B, and continues tointersect the square at point C. LetA denote (2,0).  (i) Sketchthe figure indicated in the above description.(ii) Findthe exact coordinates of A, B, and C, and label them on your sketch.(iii) Nowsuppose we have arbitrary acute angle Q  (inradians, instead of the 30O).  Again draw the sketch!(iv) Againfigure out the exact coordinates of A, B,and C and label them on your sketch.NB: You will use trig functions here!(v) Nowfigure out the equation you would have to solve to find Q to makethe area of ABCA exactly equal to thearea of the sector.  HINT: This meansarea of sector is half area of triangle. (You cannot solve such an equationexactly – this is an example of a TRANSCENDENTAL equation, so the theorems ofalgebra do not apply.)

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