Sulalgtrig7e Isg 1 7

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Sulalgtrig7e Isg 1 7

  1. 1. Mathematical N Modeling I O A T I C P L Applied A P Problems
  2. 2. SIMPLE INTEREST Interest (either made Interest rate as a or paid depending on saving or borrowing) I =Prt decimal Principal Amount Time in years (beginning amount deposited or borrowed) Suppose you borrow $1000 for 6 months at the simple interest rate of 6% per annum. What is the interest you will be charged on this loan? If you pay the loan back at the end of 6 months, what is the amount you must pay? 6% as a decimal So the interest charged is $30. 1 I = (1000 )( .06 )   = 30 You must pay back the original 2 amount borrowed plus interest time in years --- 6 months is 1/2 year so $1000 + $30 = $1030.
  3. 3. MIXTURE PROBLEMS + = Total Total Concentration Quantity Concentration Quantity or price of 1st of first + or price of 2nd of second = concentration Quantity or price How many gallons of a 25% acid Remember total quantity would be solution would you mix with 4 quantity of 1st + quantity of 2nd gallons of a 6% acid solution to We could multiply all terms by obtain a 15% acid solution? 100 to get rid of decimals. ( .25)( x ) + ( .06)( 4) = ( .15)( x + 4) x = 3.6 gal 25 x + 6( 4 ) = 15( x + 4 ) 10 x = 36
  4. 4. PHYSICS: UNIFORM MOTION You've probably heard: distance equals rate times time Using the variables from physics the equation becomes: distance s=vt time velocity (rate) These problems are easy if we just have one distance, velocity and time, but often we'll have two different situations. The best way to tackle these is to make a table with the information for each situation.
  5. 5. Uniform Motion Problem A truck traveled the first 100 miles of a trip at one speed and the last 135 miles at an average speed of 5 miles per hour less. If the entire trip took 5 hours, what was the average speed for the first part of the trip? Let's make a table with the information If you used t hours for the first part of the trip, then the distance velocity time total of 5 hours minus the t would be the time left for first part 100 v t the second part. second part 135 v-5 5-t
  6. 6. Use this formula to get an Distance = velocity x time equation for each part of trip distance velocity time first part 100 v t second part 135 v-5 5-t first part second part Solve first equation for t and substitute in second equation 100 = v t 135 = (v - 5)(5 - t) v v 100  100  135 = ( v − 5)  5 −  v  v 
  7. 7.  100  135 = ( v − 5)  5 − FOIL the right hand side   v  500 v Multiply all terms by v 135v= 5vv− 100v 25v+ − to get rid of fractions v 5v − 260v + 500 = 0 2 Combine like terms and get everything on one side Divide everything by 5 v − 52v + 100 = 0 2 ( v − 50)( v − 2) = 0 Factor or quadratic formula So v = 50 mph. v = 2 wouldn't work v = 50 or v = 2 because if you subbed in 2 for v to get velocity of second part you'd get –3.
  8. 8. WORK-RATE PROBLEM An office contains two copy machines. Machine B is known to take 12 minutes longer than Machine A to copy the company's monthly report. Using both machines together, it takes 8 minutes to reproduce the report. How long would it take each machine alone to reproduce the report? Work done by Work done by 1 complete job Machine A + Machine B = Rate for A Rate for B Time to Time to 1 over time 1 over time complete + to complete complete = 1 to complete job job alone alone 1 1 ( 8) + ( 8) = 1 t t + 12
  9. 9. 1 1 ( 8) + ( 8) = 1 t t + 12 8t(t+12)8t(t+12) t(t+12) + =1 clear equation of fractions by multiplying by common denominator t t + 12 8( t + 12 ) + 8t = t ( t + 12 ) distribute 8t + 96 + 8t = t + 12t 2 quadratic so get everything on one side = 0 t − 4t − 96 = 0 2 factor ( t − 12)( t + 8) = 0 So time for Machine A is 12 minutes and time for Machine B is 12 + 12 or 24 minutes t = 12 or t = -8 -8 doesn't make sense for time so throw it out

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