Questions and Answers
Complete the first step in solving the equation $y²=4$: y = _____
±√4
Select all solutions to the equation $y=−2$.
What is the exponential growth formula?
y = b(1 + r)^t
What is the exponential decay formula?
Signup and view all the answers
What do the variables represent in the exponential formulas?
Signup and view all the answers
What happens to the formula if the rate is 0%?
Signup and view all the answers
Predict the population of Texas in 2035 given a growth rate of 1.8% starting from 25,145,561.
Signup and view all the answers
What value should be substituted for b in the growth formula for Texas's population?
Signup and view all the answers
What is the rounded predicted population of Texas in 2035?
Signup and view all the answers
Predict the toad population 10 years after a decrease from 3800 to 3610.
Signup and view all the answers
What is the decay rate for the toad population?
Signup and view all the answers
What initial amount should be substituted for b in the toad population decay formula?
Signup and view all the answers
After 30 years, how many grams of cobalt-60 will remain from a 400 g sample?
Signup and view all the answers
What is the definition of half-life?
Signup and view all the answers
What percentage of carbon-14 levels should an archaeologist expect to find in a bone that is 10,000 years old?
Signup and view all the answers
Study Notes
Growth and Decay Equations
- The equation y²=4 yields solutions y = ±2.
- Exponential growth is modeled by the formula y = b(1 + r)ⁱ, where b is the initial amount, r is the growth rate, and t is time.
- Exponential decay follows the formula y = b(1 - r)ⁱ, highlighting a decrease over time.
Understanding Growth and Decay
- Both growth and decay formulas use similar variables: b (initial amount), r (growth/decay rate as a decimal), and t (time).
- When r is 0%, the model simplifies to a constant y = b.
Practical Application: Population Increase
- Texas population in 2010: 25,145,561, growing at 1.8% annually.
- To find the 2035 population, substitute into the growth formula using 25 for t.
- Following calculations yields a predicted population of 39,278,580.
Practical Application: Toad Population Decrease
- Toad population dropped from 3800 to 3610 in one year, indicating a decay rate of approximately 5%.
- Apply the decay formula to predict the population after 10 years, which estimates to about 2275.
Radioactive Decay Concept
- Radioactive decay is the transformation of atoms, losing particles/energy over time.
- Half-life is defined as the time required for half of a radioactive substance to decay, exemplified by cobalt-60 with a half-life of 5 years.
Assessing Remaining Substance Over Time
- For a 400 g cobalt-60 sample after 30 years, substitute into the half-life formula to find that 6.25 g remains.
Carbon-14 Decay Example
- Carbon-14 decays after an organism's death, with a half-life of 5730 years.
- For a bone estimated to be 10,000 years old, calculations determine the remaining percentage of carbon-14 present.
This outline encapsulates key concepts of growth and decay in mathematics and real-world applications, providing an efficient study guide for understanding these principles.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge of growth and decay equations in Algebra 5.08. This quiz features key equations, including the exponential growth and decay formulas and their solutions. Perfect for brushing up on essential algebra concepts.