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  1. 1. Developing Expert Voices Pre Calculus 40S enriched 2007 Hi, my name is Sandy and I like purple(: DEV PROJECT
  2. 2. Problem One <ul><li>You’re a Skydiver and you’ve just jumped out from the jet plane and you’re heading for the ground. You pull the string to the parachute and start to examine a crop circle that a farmer has created. </li></ul><ul><li>Given that the centre of the crop circle is labeled O, </li></ul><ul><li>the diameter of the circle is 10km, angle BOG is </li></ul><ul><li>120° and that D is … </li></ul><ul><li>Answer the following in radians. </li></ul><ul><li>Determine the length of the arc that subtends </li></ul><ul><li>an angle of 240°. </li></ul><ul><li>b) Determine the area of that sector using the information given above. </li></ul><ul><li>c) Determine the angle at the centre of the circle if the arc subtended by the angle is . </li></ul>DEV PROJECT
  3. 3. Problem One – answers (a) <ul><li>From the original question, the answers are supposed to be in radians. </li></ul><ul><li>The formula to change Degrees into Radians is: </li></ul><ul><li>Where D is Degrees and R is Radians. </li></ul><ul><li>Plug in the given information into the formula. It should then look like this: </li></ul><ul><li>Now you cross multiply: </li></ul><ul><li>240°( π ) = R(180°) </li></ul><ul><li>Divide everything by 180° to leave R by itself and solve! </li></ul><ul><li>The degree signs cancel each other out, and also reduces to . </li></ul><ul><li>Therefore, 240° in RADIANS is . </li></ul>DEV PROJECT
  4. 4. Problem One – answers (a) <ul><li>We have the angle in radians, so we can use that to solve for the arc length with this formula: </li></ul><ul><li>Where R is the angle in Radians, r is the radius and L is the Arc Length. </li></ul><ul><li>Now we can substitute the given information and what we found into the formula. </li></ul>DEV PROJECT The 5 in 2 π (r) was found by dividing the diameter by 2. *The radius is HALF of the diameter in a circle.
  5. 5. Problem One – answers (a) <ul><li>To get rid of the fraction in a fraction you multiply the numerator by the reciprocal of the denominator. </li></ul><ul><li>Multiply the left side and then cross multiply. </li></ul><ul><li>(4 π )(10 π ) = L(6 π ) </li></ul><ul><li>Divide everything by (6 π ) to leave L by itself. Therefore… </li></ul><ul><li>The π ’s reduce, leaving you with only 1. Also 4 x 10 = 40 all divided by 6. Reducing that, it leaves you with </li></ul>DEV PROJECT
  6. 6. Problem One – answers (a) <ul><li>A second way is to change an angle from degree to radians is: </li></ul><ul><li>Since we’ve been through this once already, it’s should be pretty straight forward on how to acquire the correct answer. You should get the same answer you got before which was: </li></ul><ul><li>So if you didn’t, you know you did something wrong. But don’t quit there, go back and try again.  </li></ul>DEV PROJECT
  7. 7. Problem One – answers (b) <ul><li>The answer again must be in radians so this is the formula you use to find the area of a sector: </li></ul><ul><li>Where Θ is the angle in radians, S is the area of the sector and r is the radius of the circle. </li></ul><ul><li>Plug in the information that we already know. </li></ul><ul><li>Again, we have to multiply the numerator by the reciprocal of the denominator. </li></ul>DEV PROJECT
  8. 8. Problem One – answers (b) <ul><li>Next, we simplify it and then it’s time to cross multiply. </li></ul><ul><li>Afterwards, we divide both sides by (6 π ) so that S is by itself. </li></ul><ul><li>We now simplify. The π ’s reduce, just like before, leaving you with only one again, and 100 ÷ 6 reduces to </li></ul>DEV PROJECT
  9. 9. Problem One – answers (b) <ul><li>Another way to solve this part of the question is to use a different formula: </li></ul><ul><li>Plugging in all the information we come up with this: </li></ul><ul><li>And from this point, it is also pretty straight forward on how to acquire the answer. Also, you should arrive at the same answer, if not… try again! </li></ul>DEV PROJECT
  10. 10. Problem One – answers (c) <ul><li>This is very similar to the first part of this question. However, instead of solving for the ARC, we’re solving for the ANGLE. </li></ul><ul><li>Where R is the angle in radians, L the Arc Length and r is the radius. </li></ul><ul><li>Now we can substitute the information we know into the formula. </li></ul>DEV PROJECT
  11. 11. Problem One – answers (c) <ul><li>Knowing the routine, we multiply the numerator by the reciprocal of the denominator again. We simplify, then cross multiply. </li></ul>DEV PROJECT
  12. 12. Problem One – answers (c) <ul><li>Once again, divide both sides by 9 π so that R will be by itself. </li></ul><ul><li>Now simplify. Remembering that the π ’s reduce you’re left with… </li></ul><ul><li>Therefore the angle subtended by the arc is . </li></ul>DEV PROJECT
  13. 13. Problem Two <ul><li>Given the graph of f(x) below, sketch the following three graphs. </li></ul><ul><li>a) b) c) </li></ul>DEV PROJECT Please note that for tests or examinations, add arrows and label your axis.
  14. 14. Problem Two – answers (a) <ul><li>The first thing you should remember before starting is: </li></ul><ul><li>STRETCHES BEFORE TRANSLATIONS </li></ul><ul><li>The basic formula for a question like this is: Af(B(x-C))+D Where A is a STRETCH (the y-coordinates are multiplied by A), B is a STRETCH (the x-coordinates are multiplied by / the reciprocal), C is a TRANSLATION (the x- coordinate is moved horizontally) and D is a TRANSLATION (the y-coordinate is moved vertically). </li></ul>DEV PROJECT
  15. 15. Problem Two – answers (a) <ul><li>Looking at the graph, we can figure out all of the coordinates of each point. </li></ul><ul><li>We’ll arrive at the numbers in this order remembering that the x-coordinate always comes before the y-coordinate. </li></ul>DEV PROJECT
  16. 16. Problem Two – answers (a) <ul><li>We can take those coordinates and apply what we’re given. </li></ul><ul><li>Af(B(x-C))+D </li></ul><ul><li>Remember that A stretches the y-coordinate and B stretches the x-coordinate. </li></ul><ul><li>First point: A(-6,-3). </li></ul><ul><li>A((-6)(3)),(-3)(-2)) </li></ul><ul><li>The 3 came from the reciprocal in the original question given above. </li></ul><ul><li>You’re then left with a new coordinate: A(-18,6). </li></ul><ul><li>BUT you’re not finished yet. </li></ul>DEV PROJECT
  17. 17. Problem Two – answers (a) <ul><li>Now you find the Translation part. </li></ul><ul><li> Af(B(x-C))+D </li></ul><ul><li>Knowing that C shifts the x-coordinate and D shifts the y-coordinate. </li></ul><ul><li>C is (-4) (even though it says x+4 in the question) because in the formula, it’s (x-C). So in order for C to be positive in the question, 4 must be a negative number. </li></ul><ul><li>(x+4)  (x-(-4)) </li></ul><ul><li>One negative, multiplied by another negative, always gives you a positive number. </li></ul><ul><li>Therefore: A((-18-4),(6+5)) </li></ul><ul><li>And the new coordinate IS: A(-22,11) </li></ul>DEV PROJECT
  18. 18. Problem Two – answers (a) <ul><li>Now there are 3 more points to do. Just for practice, try them yourself first and once your finished.. Go to the next slide and check if your answers are correct.  </li></ul><ul><li>The remaining points are: </li></ul><ul><li>B(-3,-3) </li></ul><ul><li>C(1,1) </li></ul><ul><li>D(4,1) </li></ul>DEV PROJECT
  19. 19. Problem Two – answers (a) <ul><li>Since we’ve done this once before, I’m going to go through it a bit faster. </li></ul><ul><li>B(-3,-3) </li></ul><ul><li>Stretches first!  B((-3)(3),(-3)(-2))  B(-9,6) </li></ul><ul><li>Now Translations!  B(((-9)-4),(6+5))  (-13,11) </li></ul><ul><li>C(1,1) </li></ul><ul><li>Stretches first!  C((1)(3),(1)(-2))  C(3,-2) </li></ul><ul><li>Now Translations!  C((3-4),((-2)+5))  C(-1,3) </li></ul><ul><li>D(4,1) </li></ul><ul><li>Stretches first!  D((4)(3),(1)(-2))  D(12,-2) </li></ul><ul><li>Now Translations!  D(((12)-4),(-2+5))  D(8,3) </li></ul>DEV PROJECT
  20. 20. Problem Two – answers (a) <ul><li>Now that we’ve calculated all the new coordinates, it’s time to plot those points onto a graph and connect the dots. After you’re finished, it should look like this: </li></ul><ul><li>*New Graph: Is red. </li></ul>DEV PROJECT
  21. 21. Problem Two – answers (b) <ul><li>We’re looking for the INVERSE of the function. </li></ul><ul><li>Since we already have the coordinates, all we have to do is switch the y-coordinate and the x-coordinate. </li></ul><ul><li>A(-6,-3) Now, SWITCH THEM. A(-3,-6) </li></ul><ul><li>Easy right? Okay. So lets do the others now. </li></ul><ul><li>B(-3,-3)  B(-3,-3) </li></ul><ul><li>C(1,1)  C(1,1) </li></ul><ul><li>D(4,1)  D(1,4) </li></ul>DEV PROJECT
  22. 22. Problem Two – answers (b) <ul><li>Now just like the other one, plot the points, connect the dots and you have your solution! </li></ul>DEV PROJECT
  23. 23. Problem Two – answers (c) <ul><li>It’s asking for the reciprocal of the original function. </li></ul><ul><li>Look for the “invariant points.” ((-1,-1) and (1,1)) - These points are on both the original function and the reciprocal function because the reciprocal of 1 is 1. </li></ul><ul><li>Examine the straight horizontal lines. One of them is y=(-3) and the other is y=(1). On the graph, you find the reciprocal and graph it Therefore y=(1/3) and y=(1). </li></ul>DEV PROJECT
  24. 24. Problem Two – answers (c) <ul><li>Now use the “Biggering, smallering” game. If a number increases, its reciprocal decreases and vice versa. Once it’s finished, it should look like this: </li></ul>DEV PROJECT Don’t forget to add an asymptote where the graph has zero(s) or where x = 0.
  25. 25. Problem Three <ul><li>Prove: </li></ul>DEV PROJECT
  26. 26. Problem Three - answers <ul><li>First thing to always do is to draw the “Great Wall of China.” </li></ul>DEV PROJECT Expand the problem. - We know that COT is the reciprocal of tan and instead of tan, you can use sin and cos. We also changed 1 to sinΘ/sinΘ because that equals one. So we don’t necessarily add anything to the original question.
  27. 27. Problem Three - answers DEV PROJECT Here we simply just multiplied everything out. Just like we were doing before, to get rid of the double fraction we multiplied the numerator by the reciprocal of the denominator. The sinΘ’s reduce!
  28. 28. Problem Three - answers DEV PROJECT You then end up with this after you reduce! You’re probably wondering how the heck does cos 2 Θ equal 1-sin 2 Θ? Well, knowing our identities: sin 2 Θ + cos 2 Θ = 1 If we rearrange the order, we can see that cos 2 Θ = 1-sin 2 Θ If you notice, 1-sin 2 Θ is a difference of squares. SO, it is easily factorable.
  29. 29. Problem Three - answers DEV PROJECT The 1-sinΘ’s reduce. And you’re left with 1+sinΘ! Q.E.D *When You’re finished solving, you must always put Q.E.D to indicate that you are finished.
  30. 30. Problem Four <ul><li>At this very moment, there are 1135 students that attend Daniel McIntyre High School. 2 years ago there was only 960 students that attended the High School. </li></ul><ul><li>At what rate is the student body popula tion increasing at? </li></ul><ul><li>Assuming that the rate continues to incr ease at this rate, how much longer will it take for the student body population to be 1500 </li></ul>DEV PROJECT
  31. 31. Problem Four – answers(a) <ul><li>We are looking for the rate, or the model , that the student body population is increasing. Here’s the formula that we use to solve this question: </li></ul><ul><li>P = P 0 (Model) t </li></ul><ul><li>Where P is the population at the end of the time period, P 0 is the population at the beginning of the time period, Model is the, in this case, rate at which the population increases and t is the change in time. </li></ul><ul><li>So using the information we know, we can use it to plug it into the formula. </li></ul><ul><li>1135 = 960(Model) 2 </li></ul>DEV PROJECT
  32. 32. Problem Four – answers(a) <ul><li>1135 = 960(Model) 2 </li></ul><ul><li>Divide both sides by 960 to leave (Model) 2 by itself and reduce: </li></ul><ul><li>You can use “ln” to solve or you there’s an alternate way. First way is to use ln. Take the ln of both sides: </li></ul><ul><li>Multiply by ½ on both sides to remove the 2 from the right. Calculate the left side out. </li></ul><ul><li>0.0837 = ln(Model) </li></ul><ul><li>Therefore e 0.0837 = Model. And the rate is (0.0837)(100) = 8.37% </li></ul>DEV PROJECT
  33. 33. Problem Four – answers(a) <ul><li>1135 = 960(Model) 2 </li></ul><ul><li>The other way is to do it like this: </li></ul><ul><li>And the result is 1.0873 = Model. </li></ul><ul><li>To get the actual model you minus one, because when the population increases, it keeps the original amount (1) and from there increases (+ 0.0873%). </li></ul>DEV PROJECT These reduce.
  34. 34. Problem Four – answers(b) <ul><li>We’re asked to find the time it takes for the population to increase from 1135 to 1500. </li></ul><ul><li>1500 = 1135(1.0873) t </li></ul><ul><li>Divide both sides by 1135 and reduce: </li></ul>DEV PROJECT
  35. 35. Problem Four – answers(b) <ul><li>Now you can use ln to solve for t: </li></ul><ul><li>Divide both sides by ln(1.0873). </li></ul><ul><li>Then solve for t! and t is 3.331428783  3.3314 years. </li></ul><ul><li>So it will take 3.3314 years for the student body population to grow to 1500. </li></ul>DEV PROJECT
  36. 36. Toodle-Loo Kangaroo! =) DEV PROJECT