Question 3 Solution
- 1. Exponential Modeling (Exponential Growth) Exponents and Logarithms Question 3 Solution
- 2. S o l u t i o n t o ( a ) :
- 3. Because of the little information given in the question, we use this equation: A = A o ( model ) t is the original amount of "substance" at the beginning of the time period. In this case, the "substance" is the population. A o A is the final amount of "substance" at the end of the time period. model is our model for the growth (or decay) of the substance, it is usually an exponential expression in base 10 or base e, although any base can be used. t is the amount of time that has passed for the substance to grow (or decay) from A o to A .
- 4. A = A o ( model ) t Now that we have our equation and we know what each variable is for, we can plug each one in and search for our model. A o = 472 A = 609 model : this is what we'll be seaching for as we solve this question. Δ t = 17 - 0 t = 17 609 = 472 (model) 17 (model) 17 = There are two ways to calculate the model of this question base e base 10 609 472
- 5. base e (model) 17 = First you take the ln of both sides. When you take the ln (The "Natural Log") of both sides, they have the same base e. ln ln = (model) 17 Now using the "Power Law", bring down the exponent so that it is in front of the logarithm. ln ln = (model) 17 Now multiply on both sides so that ln(model) will be isolated. ln ln = (model) 17 Calculate what is on the left side. Don't forget about brackets when you enter it. Round to 4 decimal places, but store your answer. 0.01499 = ln (model) Take the base e of both sides and you have your model. e 0.01499 = model This is what A would equal to when t is 0. 609 472 609 472 ( ) 609 472 ( ) 609 472 ( ) 1 17 1 17 ( ) 1 17 ( ) A = 472 ( e 0.01499 ) t
- 6. base 10 (model) 17 = First you take the log of both sides. When you take the log (The "Common Log") of both sides, they have the same base 10. log log = (model) 17 Now using the "Power Law", bring down the exponent so that it is in front of the logarithm. log log = (model) 17 Now multiply on both sides so that log(model) will be isolated. log log = (model) 17 Calculate what is on the left side. Don't forget about brackets when you enter it. Round to 4 decimal places, but store your answer. 0.0065 = log (model) Take the base 10 of both sides and you have your model. 10 0.0065 = model 609 472 609 472 ( ) 609 472 ( ) 609 472 ( ) 1 17 1 17 ( ) This is what A would equal to when t is 0. A = 472 ( 10 0.0065 ) t 1 17 ( )
- 7. Our models showing the population growth would be... Both can be used to calculate the exponential growth or decay of the population. A = 472 ( 10 0.0065 ) t A = 472 ( e 0.01499 ) t
- 8. S o l u t i o n t o ( b ) :
- 9. Using any of the equations from the previous question, we can calculate the time it will take A o = 472 A = 851 t = ? 851 = 472 ( e 0.01499 ) t e 0.01499 t = Now like in the previous question, make both sides the same base. Take the "natural log" of both sides. ln 0.01499 t = ln 0.01499 t = ln Multiply both sides by to isolate the t. Calculate to get the approxiamate time. exact value: t = ln 39.3203 ≈ t 851 472 851 472 1 0.01499 ( ) 851 472 1 0.01499 ( ) 1 0.01499 1 0.01499 ( ) 851 472
- 10. For the population to reach 851 people, it would take approxiamately: 39 years