Exponential Decay Formula

Exponential Decay Formula: A quantity is said to be in exponential decay if it decreases at a rate proportional to its current value. In exponential decay, a quantity drops slowly at first before rapidly decreasing. The exponential decay formula is used to calculate population decay (depreciation), and it can also be used to calculate half-life (the amount of time for the population to become half of its size).

In this article, we have provided the formula for Exponential Decay, along with some examples of it.

What is the Exponential Decay Formula?

The exponential decay formula is used to model situations where a quantity decreases at a rate proportional to its current value. It’s a common formula in physics, chemistry, finance, and other fields for describing processes such as radioactive decay, cooling, and depreciation of assets.

Exponential Decay Formula

In exponential decay, the original amount decreases by the same percent over some time. A variation of the growth equation can be used as the general equation for exponential decay.

The formula for exponential decay is as follows:

y = a(1 – r)^t

where a is initial amount, t is time, y is the final amount and r is the rate of decay.

Solved Examples on Exponential Decay Formula

Problem 1. Every day, a fully inflated child’s pool raft loses 6.6 percent of its air. 4500 cubic inches of air were originally stored in the raft. To indicate the loss of air, write an equation.

Solution:

The equation for exponential decay is y = a(1 – r)t.

Here, a = 4500, r = 6.6% or 0.066

Hence, y = 4500(1 – 0.066)^t

⇒ y = 4500(0.934)^t

Here y is the air in the raft in cubic inches after t days.

Problem 2. Find the amount of air in the raft after 7 days in the above problem.

Solution:

As per the above problem, y = 4500(0.934)^t.

Here, t = 7. Then,

⇒ y = 4500(0.934)⁷

⇒ y ≈ 2790

Problem 3. A town’s population has been declining at a pace of around 0.3 percent per year on average. The population was 88647 in 2000. Create a formula to reflect the population since the year 2000.

Solution:

The equation for exponential decay is y = a(1 – r)^t.

Here, a = 88647, r = 0.3% or 0.003

Hence, y = 88647(1 – 0.003)^t

⇒ y = 88647(0.997)^t

Problem 4. Find the population of the above town in 2010 if the trend continues.

Solution:

As per the above problem, y = 88647(0.997)^t

Here, t = 2010 – 2000 = 10. Then,

⇒ y =  88647(0.997)¹⁰

⇒ y ≈ 86024 people

Problem 5. An investment of $4500 has been losing value at 2.5% annually. Write an equation to represent its value in t years.

Solution:

The equation for exponential decay is y = a(1 – r)^t.

Here, a = 4500, r = 2.5% or 0.025

Hence, y = 4500(1 – 0.025)^t

⇒ y = 4500(0.975)^t

Problem 6. Find the value of the investment in 5 years for the above problem.

Solution:

As per the above problem, y = 4500(0.975)^t.

Here, t = 5. Then,

y = 4500(0.975)⁵

⇒ y = $3964.93

or, y ≈ $3965

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Summary – Exponential Decay Formula

The exponential decay formula is a crucial mathematical model used to describe the process where a quantity diminishes over time at a rate directly proportional to its current value. This formula is extensively applied across various domains, including physics for radioactive decay, chemistry for reaction rates, and finance for asset depreciation. It encapsulates how an initial quantity, denoted by a, diminishes by a consistent percentage over equal time intervals, leading to a rapid decrease after a slower initial decline. Mathematically, the exponential decay is expressed as y=a(1−r)t, where y represents the quantity after time t, a is the initial amount, r is the decay rate, and t is the elapsed time. This formula is instrumental in calculating not just the depreciation of values over time but also in determining half-lives, which is the time required for a quantity to reduce to half its initial value.

Practice Problems on Exponential Decay Formula

1. A sample of a radioactive isotope has an initial mass of 20 grams. The half-life of the isotope is 3 years. Calculate the remaining mass of the isotope after 9 years.

2. A new car is purchased for $25,000. The car depreciates at a rate of 20% per year. How much will the car be worth after 5 years?

3. A species of fish in a lake is decreasing at a rate of 10% per year due to pollution. If the current population is 5000 fish, what will be the population after 2 years?

4. A chemical solution is degrading at a rate of 15% per hour. If you start with 100 milliliters of the solution, how much of the original chemical will remain after 6 hours?

5. A scientist has 160 grams of a substance that decays exponentially. After 4 hours, only 20 grams of the substance remain. What is the half-life of the substance?

FAQs on Exponential Decay Formula

What is Exponential Decay Formula?

The Exponential Decay Formula is used to describe a process where a quantity decreases over time at a rate proportional to its current value.

What is the Formula For Finding the Exponential Decay?

The formula for finding exponential decay is given by: y=ae − rt where:

• y is the quantity at time t,
• a is the initial quantity (at t=0),
• r is the decay rate,
• t is the time,
• e is the base of the natural logarithm, approximately equal to 2.71828.