Exponential & Logarithmic Models — Challenge
Growth and decay that depend on how much is already there, and the logs that solve them.
What these problems cover
These questions use 5 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
- Factor — A number that divides evenly into another.
- Growth factor — b itself. b > 1 means growth, 0 < b < 1 means decay.
- Half-life — The time for a decaying quantity to fall to half.
- Magnitude — The length of the vector, written |v|.
Problem 1
If y=3· 2x, what is y when x=0?
- 0
- 1
- 2
- 3
- 6
Worked answer
3
Any non-zero number to the power 0 is 1, so 20 = 1.
Then y = 3· 1 = 3.
In general the starting value of y = a· bx is a.
Problem 2
A car worth $20,000 loses 10% of its value each year. What is it worth in dollars after one year?
- $17,000
- $18,000
- $19,000
- $19,800
- $22,000
Worked answer
$18,000
Losing 10% leaves 90% of the value, so multiply by 0.90.
20,000· 0.90 = 18,000 dollars.
Problem 3
A population of 200 triples every decade. What is it after two decades?
- 400
- 600
- 1200
- 1800
- 3600
Worked answer
1800
Tripling twice multiplies by 32 = 9.
200· 9 = 1800.
Multiplying by 3 and then adding another 3 times the start would give 1200, which treats the growth as additive.
Problem 4
An account holds $500 and earns 5% interest a year, compounded annually. What is the balance in dollars after two years?
- $525.00
- $550.00
- $551.25
- $552.50
- $605.00
Worked answer
$551.25
Each year multiplies the balance by 1.05, so two years multiply by 1.052 = 1.1025.
500· 1.1025 = 551.25 dollars.
Simple interest would give 550, missing the 1.25 of interest earned on the first year's interest.
Problem 5
A quantity halves every 3 hours. What fraction of it remains after 12 hours?
- 14
- 18
- 116
- 112
- 132
Worked answer
116
Twelve hours contains 123 = 4 halving periods.
Halving four times leaves (12)4 = 116.
Dividing by 4 to get 14 would treat the decay as linear.
Problem 6
The model y=100(1.2)x describes growth at what rate per step?
- 20%
- 1.2%
- 120%
- 80%
- 12%
Worked answer
20%
A base of 1.2 means each step multiplies the quantity by 1.2.
Multiplying by 1.2 is the same as keeping 100% and adding another 20%.
So the growth rate is 20% per step. A base of 0.8 would instead mean a 20% decline.
Problem 7
A bacteria culture doubles every 20 minutes. If there are 400 cells now, how many will there be in 2 hours?
- 2400
- 4800
- 12800
- 25600
- 51200
Worked answer
25600
Two hours is 120 minutes, which is 120 ÷ 20 = 6 doublings. Six doublings multiply by 26=64. 400 × 64 = 25600. Adding 400 six times would give only 2800 — doubling is not repeated addition.
Problem 8
A substance decays to 18 of its original amount in 12 years. What is its half-life?
- 2 years
- 3 years
- 4 years
- 6 years
- 8 years
Worked answer
4 years
18=(12)3, so three half-lives have passed. Three half-lives take 12 years. So one half-life is 12 ÷ 3 = 4 years.