Exponential Equations Applications

Learning Outcomes: To determine the solution of exponential equations in which the bases are powers of one another To solve problems that involve exponential growth or decay To solve problems that involve the application of exponential equations to loans, mortgages, and investments.

A cell doubles every 4 min. If there are 500 cells originally, Exponential Growth A cell doubles every 4 min. If there are 500 cells originally, how long would it take to reach 16 000 cells? N(t) Number of bacteria after t minutes No Number of bacteria originally t Time passed d Doubling time t = 20 Therefore, it would take 20 min for the cells to reach 16 000.

How long does it take for the cells to divide to 4096? http://www.google.ca/imgres?imgurl=http://www.maine.gov/dhhs/mecdc/environmental-health/water/images/beakers.jpg&imgrefurl=http://www.maine.gov/dhhs/mecdc/environmental-health/water/dwp-services/labcert/labcert.htm&h=540&w=632&sz=35&tbnid=yaskwiD900Ef1M:&tbnh=89&tbnw=104&prev=/search%3Fq%3Dlaboratory%2Bimages%26tbm%3Disch%26tbo%3Du&zoom=1&q=laboratory+images&usg=__5DLWd9MQe9ZtTe6hcMJMALh2F20=&docid=cTYsDeSCPYOFOM&hl=en&sa=X&ei=jZd0UOiMNcipiALPiYFI&ved=0CCkQ9QEwAw&dur=156 Ex) A lab tech placed a bacterial cell into a vial at 5 am. The cells divide in such a way that the number of cells doubles every 4 minutes. The vial is full one hour later. How long does it take for the cells to divide to 4096? Solution: Construct equation.

b. At what time is the vial half full? Solution 1: If it was full an hour after the start, 5am then it is full at 6 am. It would have to be half full one time interval before, (4 minutes before) 5:56 am Solution 2: A = half full This answer means 4 minutes prior to being full. 5:56 am

Solution: c. When is the vial 1/16 full? This means 16 minutes before it was full, which is 5:44am.

Compound Interest: The formula which can be used to calculate compound interest is where, A represents the final amount P represents the initial amount i represents the interest rate per compounding period n represents the number of compounding periods

Note: i does NOT always represent the annual interest rate n does NOT always represent the number of years

Ex. $7000 is invested in a 6 year GIC compounded quarterly at a rate of 5% per annum. Determine the value of the investment at the end of the term. A = ? P = 7000 i = n = (6)(4) = 24

Remember that the rate and compounding periods change: Annually: keep same Semi-annually: divide rate by 2, double compounding period Quarterly: divide rate by 4, multiply compounding period by 4 Monthly: divide rate by 12, multiply compounding period by 12

Ex. Determine how long $1000 needs to be invested in an account that earns 8.3% compounded semi-annually before it increases in value to $1490. Isolate the power and then solve for the exponent using guess and check or use technology to find the intersection point.

The half-life of sodium 24 is 15 h. How long would it take Exponential Decay The half-life of sodium 24 is 15 h. How long would it take for 1600 mg to decay to 100 mg? A(t) Amount after a given period of time Ao Amount originally t Time passed h Half-life t = 60 It would take 60 h to decay to 100 mg.

Write the equation to represent this situation. Ex) In April 1986, the nuclear accident at Chernobyl contaminated the atmosphere with radioactive iodine-131. If the half life of this substance is 8 days, determine the number of days to the nearest day, that it took for the level of radiation to reduce to 2% of the original level. Write the equation to represent this situation. Since we cannot make these the same base, use technology to solve. y1 = 0.02 y2 = 0.05^(x/8) x = 45.15 = 45 days

Assignment p. 344 #10, 14, p. 356 #12 p. 364 #8 – 14, C2