Exponential Decay

Show your work for each problem.

1.
Which function shows exponential decay?
1. a) $y = 40 \cdot 0.75^t$
2. b) $y = 40 \cdot 1.2^t$
3. c) $y = 40 \cdot 3^t$
4. d) $y = 40 + 0.75t$
Answer: $y = 40 \cdot 0.75^t$
In $y = 40 \cdot 0.75^t$ the base is $0.75$ , and $0 < 0.75 < 1$ , so it is decay ( $25\%$ per period). $1.2$ and $3$ are bases of growth, and $40 + 0.75t$ is linear, not exponential.
2.
A quantity decreases by $15\%$ every year. What is the decay factor $b$ ? Write it as a decimal.
Answer: 0.85
$b = 1 - 0.15 = 0.85$ . Each year $85\%$ of the quantity is kept.
3.
For $f(t) = 500 \cdot 0.6^t$ , by what percent does $f$ decrease each period?
1. a)6%
2. b)60%
3. c)0.4%
4. d)40%
Answer: 40%
$r = 1 - 0.6 = 0.4 = 40\%$ . The function keeps $60\%$ and loses $40\%$ each period.
4.
Function 1 is $y = 150 \cdot 1.04^t$ and function 2 is $y = 150 \cdot 0.96^t$ . Which one shows decay?
Answer: 2
1. What is the base of function 1? 1.04
2. What is the base of function 2? 0.96
3. By what percent does function 2 decrease each period? Type the number only. 4
4. Which function shows decay? Type 1 or 2. 2
5.
An exponential model has decay factor $0.93$ . What is the decay rate in percent? Type the number only (for example 5 for $5\%$ ).
Answer: 7
$r = 1 - 0.93 = 0.07 = 7\%$ . The quantity loses $7\%$ per period.
6.
A laptop costs 1000 dollars and loses $20\%$ of its value each year. What is it worth after 2 years (in dollars)?
Answer: 640
1. What is the decay factor? 0.8
2. Calculate $0.8^2$ 0.64
3. Multiply: $1000 \cdot 0.64$ 640
7.
A phone costs 600 dollars and loses $20\%$ of its value each year. Which expression gives its value after $t$ years?
1. a) $600 \cdot 1.2^t$
2. b) $600 \cdot 0.8^t$
3. c) $600 - 0.2t$
4. d) $600 \cdot 0.2^t$
Answer: $600 \cdot 0.8^t$
Losing $20\%$ means keeping $80\%$ , so the value is $600 \cdot 0.8^t$ . $0.2^t$ would keep only $20\%$ , $1.2^t$ is growth, and $600 - 0.2t$ is linear.
8.
Analyze the model $f(t) = 2000 \cdot 0.7^t$ and find $f(2)$ .
Answer: 980
1. What is the initial value? 2000
2. What is the decay factor? 0.7
3. What is the decay rate in percent? Type the number only. 30
4. Calculate $f(2)$ 980
9.
A quantity loses $20\%$ in every period. What percent of the starting amount is left after 4 periods? Type the number only, with two decimal places.
Answer: 40.96
$0.8^4 = 0.4096$ , so $40.96\%$ of the starting amount is left, whatever the starting amount is.
10.
A car is bought for 20000 dollars and loses $10\%$ of its value each year. What is it worth after 3 years (in dollars)?
Answer: 14580
1. What is the decay factor? 0.9
2. Calculate $0.9^3$ 0.729
3. Multiply: $20000 \cdot 0.729$ 14580
11.
Which statement about $y = 80 \cdot 0.75^t$ is NOT true?
1. a)The initial value is 80
2. b)It reaches 0 after 4 periods
3. c)After one period, y = 60
4. d)It decreases by 25% each period
Answer: It reaches 0 after 4 periods
The initial value is $80$ , the rate is $1 - 0.75 = 25\%$ , and $80 \cdot 0.75 = 60$ . But $80 \cdot 0.75^4 \approx 25.31$ , not $0$ : an exponential decay never reaches zero.
12.
A patient takes 500 mg of a medicine. The body removes $10\%$ of the remaining amount every hour. How many milligrams are left after 5 hours? Round to the nearest whole milligram.
Answer: 295
1. What is the decay factor? 0.9
2. Calculate $0.9^5$ , rounded to four decimal places 0.5905
3. Multiply by 500 and round to the nearest whole milligram 295
13.
A machine is worth 10000 dollars. Model A: it loses 1500 dollars per year. Model B: it loses $15\%$ per year. Which model gives the higher value after 5 years?
1. a)Model B (about 4437)
2. b)Both give the same value
3. c)Model A (2500)
4. d)Both are worth 0
Answer: Model B (about 4437)
Model A: $10000 - 7500 = 2500$ . Model B: $10000 \cdot 0.85^5 \approx 10000 \cdot 0.4437 \approx 4437$ . The exponential model is higher, because its yearly losses shrink.
14.
Coffee is $85°C$ in a room at $20°C$ . In a simple model, the difference between the coffee and the room shrinks by $10\%$ every minute. What is the coffee's temperature after 3 minutes, in °C? Round to one decimal place.
Answer: 67.4
Difference: $65 \cdot 0.9^3 = 65 \cdot 0.729 = 47.385$ . Temperature: $20 + 47.385 = 67.385 \approx 67.4°C$ .
15.
A machine costs 12000 dollars. Plan A: it loses 1500 dollars per year. Plan B: it loses $20\%$ per year. After 4 years, how many dollars higher is the value under plan A than under plan B?
Answer: 1084.8
1. What is the value under plan A after 4 years? 6000
2. Calculate $0.8^4$ 0.4096
3. What is the value under plan B after 4 years? 4915.2
4. How much higher is plan A? 1084.8
16.
A bicycle's value drops from 400 to 324 in 2 years, losing the same percent each year. What is the yearly percent decrease? Type the number only.
Answer: 10
$b^2 = \frac{324}{400} = 0.81$ , so $b = 0.9$ . The decay rate is $1 - 0.9 = 0.1 = 10\%$ . Check: $400 \to 360 \to 324$ .
17.
A scooter costs 1000 dollars and loses $20\%$ of its value each year. After how many full years is it worth less than 450 dollars for the first time?
Answer: 4
1. Value after 1 year? 800
2. Value after 2 years? 640
3. Value after 3 years? 512
4. Value after 4 years? 409.6
5. After how many full years is the value below 450 for the first time? 4
18.
A patient receives 800 mg of a medicine. Each hour the body removes $40\%$ of the amount still present. How many milligrams are left after 3 hours, and what percent of the dose is that?
Answer: 172.8
1. What is the decay factor? 0.6
2. Calculate $0.6^3$ 0.216
3. How many milligrams are left? 172.8
4. What percent of the dose is left? Type the number only. 21.6