Exponential Function Text

Question

An element with a mass of 680 grams decays by 12.1% per minute.To the nearest minute,how long will it be until there are 230 grams of the element remaining?

Answer 1

Hi Latifah.

Remember that the formula for exponential decay doesn't keep track of what was lost, but HOW MUCH REMAINS!

The basic formula for decay is A(t) = A₀(1 - r)^t, where A_o is the initial amount r is the decay rate (CONVERTED TO A DECIMAL!) and t counts decay periods.

In this problem we have an initial amount of 680 grams. Each minute 12.1% is lost, and 87.9% remains. The exponent t is counting the minutes. We're asked to find how long it takes until only 230 grams left.

I can think of two methods to solve this problem, both require a calculator.

Method 1. Use a recursive (repetitive) formula.

Enter 680 into a calculator and hit enter or =.

Next enter *.879 and hit enter (keeping count as you go) until the answer is less than 230.

Below is what came up on my calculator each time I hit the "Enter" button.

1. 597.72
2. 525.40
3. 461.82
4. 405.94
5. 356.82
6. 313.65
7. 275.70
8. 242.34
9. 213.01

It takes between 8 and 9 minutes.

Method 2. Use Logarithms

We start by setting the exponential expression equal to the specified amount, getting;

230 = 680(.879)^t , divide by 680

23/68 = .879^t, Take the log_.879 of each side

log_.879(23/68) = t

t ≈ 8.41 minutes

If your calculator doesn't allow you to specify the index, use the Change of Base formula.

t = log(23/68)/log(.879), you'll get the same answer.

BE SURE TO CLOSE THE PARENTHESES BEFORE DIVIDING!

Answer 2

N/N₀ = (1 - .121)^t = .879^t

Solving for t by taking ln of both sides

ln(N/N₀) = ln(.879^t) = t*ln(.879)

t = ln(230/680)/ln(.879) = 8.4 min.

You have to decide whether they just want you to round or they want the minutes to be at the fraction required or less.

Answer 3