Continuous Growth and Decay

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Show your work for each problem.

1.
A substance decays continuously at a rate of 4% per year. Which formula models the amount $A$ after $t$ years, starting from $P$ ?
1. a) $A = Pe^{-0.04t}$
2. b) $A = P(1.04)^t$
3. c) $A = Pe^{0.04t}$
4. d) $A = Pe^{-4t}$
2.
In the model $A = 250e^{0.07t}$ , what is the initial amount?
1. a) $0.07$
2. b) $e$
3. c) $250$
4. d) $267.5$
3.
Use $A = Pe^{rt}$ with $P = 1000$ , $r = 0.04$ and $t = 5$ . Use $e^{0.2} \approx 1.2214$ . What is $A$ ? Round to one decimal place.
4.
A population follows $N = 300e^{0.12t}$ , with $t$ in years. What is the continuous growth rate as a percent? Write only the number, without the % sign.
5.
You invest 1500 dollars at 3% compounded continuously for 10 years. Use $e^{0.3} \approx 1.3499$ . How much do you have at the end? Round to the nearest dollar.
6.
A culture of 400 bacteria grows continuously at 25% per hour. How many bacteria are there after 4 hours? Use $e \approx 2.71828$ and round to the nearest whole number.
7.
What does the model $N = 50e^{-0.3t}$ describe?
1. a)Continuous growth at 0.3%
2. b)Continuous decay at 30%
3. c)Continuous decay at 3%
4. d)Continuous growth at 30%
8.
Two accounts start with the same amount, the same interest rate and the same time. One compounds annually, the other continuously. Which one ends with more money?
1. a)The annual one
2. b)The continuous one
3. c)They end with the same amount
4. d)It depends on the starting amount
9.
200 grams of a radioactive substance decay continuously at a rate of 5% per year. Use $e^{-0.5} \approx 0.6065$ . How many grams are left after 10 years? Round to one decimal place.
10.
How much must you deposit now, at 5% compounded continuously, to have 10000 dollars after 10 years? Use $e^{0.5} \approx 1.6487$ and round to the nearest dollar.
11.
A patient receives 60 mg of a medicine. The body removes it continuously at a rate of 20% per hour. Use $e^{-0.6} \approx 0.5488$ . How many mg have been removed after 3 hours? Round to one decimal place.
12.
You invest 3000 dollars at 4% for 5 years. Use $1.04^{5} \approx 1.2167$ and $e^{0.2} \approx 1.2214$ . Compare annual and continuous compounding. Round each amount to the nearest dollar.
13.
How much must you deposit today, at 4% compounded continuously, to have 6000 dollars after 5 years? Use the calculator ( $e \approx 2.71828$ ) and round the final answer to the nearest dollar.
14.
Offer A: 6% compounded annually. Offer B: 5.85% compounded continuously. Use $e^{0.0585} \approx 1.0602$ . Which offer gives more money after one year?
1. a)It cannot be decided without the deposit
2. b)Both give the same
3. c)Offer B
4. d)Offer A
15.
5000 dollars are invested at 8% for 10 years. Use $1.08^{10} \approx 2.158925$ and $e^{0.8} \approx 2.225541$ . How many dollars more does continuous compounding give than annual compounding? Round to the nearest dollar.
16.
You want to invest 2000 dollars for 10 years. Bank A pays 5% compounded annually, Bank B pays 4.9% compounded continuously. Use $1.05^{10} \approx 1.6289$ and $e^{0.49} \approx 1.6323$ . Round each amount to the nearest dollar. How many dollars more does the better bank give?
17.
A sample of 500 grams of a radioactive substance decays continuously at 2% per year. Use $e^{-1} \approx 0.3679$ . How many grams are left after 50 years? Round to the nearest gram.
18.
A culture starts with 1000 bacteria and grows continuously at 30% per hour. Use $e^{0.6} \approx 1.8221$ and $e^{1.2} \approx 3.3201$ . By what factor does the culture grow from hour 2 to hour 4? First find the number of bacteria after 2 hours and after 4 hours (nearest whole number), then give the factor to two decimal places.