Algebra 5.08: Growth and Decay Equations

Complete the first step in solving the equation $y²=4$: y = _____

Select all solutions to the equation $y=−2$.

What is the exponential growth formula?

What is the exponential decay formula?

What do the variables represent in the exponential formulas?

What happens to the formula if the rate is 0%?

Predict the population of Texas in 2035 given a growth rate of 1.8% starting from 25,145,561.

What value should be substituted for b in the growth formula for Texas's population?

What is the rounded predicted population of Texas in 2035?

Predict the toad population 10 years after a decrease from 3800 to 3610.

What is the decay rate for the toad population?

What initial amount should be substituted for b in the toad population decay formula?

After 30 years, how many grams of cobalt-60 will remain from a 400 g sample?

What is the definition of half-life?

What percentage of carbon-14 levels should an archaeologist expect to find in a bone that is 10,000 years old?

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.

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.