Describe one or two situations outside of school where a pattern of linear growth, exponential growth or exponential decay exist and would be useful.

Answer 1

Linear growth can be used to explain many situations where rate of growth is constant. For example, if I start out with a checking account of $0.00, and deposit a weekly allowance of $100, my checking account is consistently growing by $100 every week (given the condition that I never stop putting $100/week in and never withdrawing any money). I can use the equation y = 100x, y being the total amount in my checking account and x being the number of weeks starting from the first weekly deposit, to calculate how much money I will have after x weeks.

Using a similar example, if I instead put those $100 in a savings account, I can incur interest. Most if not all savings accounts offer a fixed interest rate on your account balance, and its growth can be modeled exponentially. Interest rates are usually expressed in a fixed APY, or annual percentage yield. For instance, if I decide to open a savings account with B Bank, and B Bank offers an annual percentage yield (APY) of 1% (which is actually pretty high believe it or not), that means every year I will earn 1% of my savings account balance. If the principal, or starting, amount is once again $100, and I don’t put anything into our withdraw any money from my savings, I will have earned 1% of $100, or $1. Let’s then suppose that this account remains untouched. The next year the interest I will earn off that money will then be 1% of $101, since I earned an extra $1 from the previous year ($100 + $1). In this way, exponential growth is compounding, meaning that the new amount is calculated based off a combination of the original amount plus what was previously accumulated. For the above example we would use the equation y=100(1 + 0.01)^x, y being total amount of $ and x being time, or number of years starting from when the account was first opened.

Exponential decay is present in physical processes such as radioactive decay, where radioactive atoms have an unstable nuclei and emit charged particles and energy. The amount of time it takes for half of the radioactive atoms to decay is called the “half-life” of the element. For example, Carbon-14 (an isotope, or variant, of carbon), has a half-life of 5730 years. If we start with 4 carbon-14 atoms, then after one half-life, or 5730 yrs, we should expect to find 2 remaining carbon-14 atom. Subsequently, after two half-lives we should expect to find 50% less, or one carbon-14. Such a process can be expressed using the equation for exponential decay or y=A(1-r)^x where A is the original amount of carbon-14, r is the rate of decay and x is the number of half lives that have passed. In this case the rate of decay would be 50%, or 0.5 (because after each half-life the number of atoms is 1/2 less, or reduced by 50%).

Hope these examples help :)