An element with mass 730 grams decays by 27.6% per minute. How much of the element is remaining after 12 minutes, to the nearest 10th of a gram?

Question

Exponential Functions - Basic

Answer 1

the underlying formula you want to use here is

A = P(1 - r)^t

for this set of parameters that becomes

A = 730(1 - 0.276)^12

A = 730(0.724)^12

A = 15.0284

meaning that approximately 15.0 grams of the element will be remaining after 12 minutes

Answer 2

Let A(t) = mass remaining after t minutes

A(t) = 730(1 - 0.276)^t = 730(0.724)^t

A(12) = 730(0.724)¹² ≈ 15.1 grams

Answer 3

Typically, when you are struggling with problems of a type, such as decay problems, it indicates a lack of understanding of the concepts behind the formulas we use. Explaining how to identify values from the question and substitute them into a memorized formula won't solve this problem, nor will it help when you forget a formula on test day.

A very effective approach in many situations is to simplify the problem. Use nice numbers and work through the problem step by step, looking for patterns that will help you grasp the formula.

Let's say we have 1000 grams with a 10% decay rate per period:

1. After 1 period: 90% remains or 1000 * 0.9 = 900 grams
2. After 2 periods: 90% of 900 remains or 900 * 0.9 = 810 grams
3. After 3 periods: 90% of 810 remains or 810 * 0.9 = 729

Notice the pattern: 1000 * 0.9 * 0.9 * 0.9 = 1000(0.9)³

This reveals the core concept: with each period, we multiply by the percentage that remains, not the percentage that decays.

Applying this to our original problem:

1. Starting amount: 730 grams
2. Percentage remaining after each minute: 100% - 27.6% = 72.4% or 1 - 0.276 = 0.724
3. Number of minutes: 12

Solution: 730(0.724)¹² = 15.1 grams

By working through simpler numbers first, you can rediscover the formula anytime:

Starting Amount * (Percentage Remaining)^{(Number of time periods)}

Bonus: Working through problems in practice in this manner deepens your understanding of the formula so you can use it more confidently in the future to save time.

Answer 4

With exponential functions, you will have a rate of increase or decrease. We can call that rate (r). We will also have a period, the amount of times that this rate (r) will affect the equation. We can call this period (n).

So, in this problem, our "r" is a decay of 27.6% per minute and our "n" is 12 periods or 12 minutes. Lastly, we have our starting value, let's call it "m" for mass. Using the basic exponential function formula we can set up this problem as a function of time.

f(n) = m * (1 - r)ⁿ

**Remember, since the element is decaying, we will subtract the rate of change from 1.**

f(12) = (730 grams) * (1 - 0.276)¹²

f(12) = (730 grams) * (0.724)¹²

f(12) = 15.1 grams

**So, after 12 minutes, the mass of the element is approximately 15.1 grams**