Growth and decay · Grade 12 · 4 units
In Growth and decay (Grade 12 · 4 units) you practise 5 skills: The growth and decay model · One time unit · Finding amounts and times · Half life · Growth and decay on a graph. In growth or decay, an amount changes in the same ratio at every step. If it grows by 5 percent each year, it is multiplied by 1.05 each year; if it falls by 5 percent, it is multiplied by 0.95. The number it is multiplied by is called the factor.
The amount after several steps is the starting amount times the factor to the power of the number of steps. For example, 20,000 that grows by 4 percent each year is about 24,333 after five years. When a step is longer than the time unit, count how many steps fit in the time given: an amount that doubles every 3 hours is multiplied by 16 in 12 hours. The half life is the time it takes an amount to halve. To find a time, look for the number of steps that gives the amount. When no whole number of steps gives it exactly, look for the first whole step that passes it.
Tip: Growth is multiplying by a factor greater than 1, and decay is multiplying by a factor between 0 and 1.
What you learn
The growth and decay model
For example: The number of trees in a forest after t years is f(t) = f(0) × 0.95^t . By how many percent does the number of trees in a forest fall each year?
Answer and solution
The answer: 5
The factor is less than 1, and the fall at each step is 1 − 0.95 = 0.05 , that is 5 percent.
One time unit
For example: The number of bacteria in a dish doubles every 6 hours. By what number is it multiplied over 12 hours?
Answer and solution
The answer: 4
There are 2 steps of 6 hours in 12 hours, so: 2² = 4
Finding amounts and times
For example: Today the number of bacteria in a dish is 1,100. Each hour the number of bacteria in a dish is multiplied by 2. In how many hours will the number of bacteria in a dish be 35,200?
Answer and solution
The answer: 5
Call the number of hours t : 1,100 × 2^t = 35,200 . Divide by the starting amount: 2^t = 32 , and 2⁵ = 32 . So t = 5 .
Half life
For example: Today the mass of a radioactive sample is 120 grams. The half life of the substance is 8 days. What will the mass of a radioactive sample be in 16 days?
Answer and solution
The answer: 30 grams
There are 2 half lives in 16 days, so: 120 × (½)² = 30
Growth and decay on a graph
For example: The graph marks the number of bacteria in a dish after whole hours. Each hour the number is multiplied by 2.
What was the number at the start?Answer and solution
The answer: 115
The start is one hour before the first point, so divide the first value by 2: 230 ÷ 2 = 115
Not on the site yet
Parts of this topic that are not built yet: A time that is not a whole number of steps, by a logarithm.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).