Question 11·Medium·Nonlinear Functions
A radioactive substance loses 13% of its mass each hour. The substance has an initial mass of grams.
Which expression gives the mass, , in grams remaining after hours?
For percent growth or decay word problems, first convert the percent change to a decimal and decide what fraction of the quantity remains after one time step: for a loss of , the multiplier is . Then write an exponential model of the form for decay (or for growth). Quickly eliminate linear expressions (which subtract or add a constant each step) and exponential expressions whose base is greater than 1 when the situation clearly describes decay.
Hints
Think about what remains, not what is lost
If the substance loses 13% of its mass each hour, what percent of its mass remains after each hour? Convert that percent to a decimal.
Decide between subtraction and multiplication
Ask yourself: are we subtracting the same amount of grams every hour, or are we keeping the same percentage of the current mass every hour? That tells you whether the model should be linear or exponential.
Use repeated multiplication for hours
Once you know what fraction remains after 1 hour, think about what happens after 2, 3, and then hours. How do you represent repeated multiplication of the same factor times?
Check whether it’s growth or decay
For exponential functions, a base bigger than 1 means growth, and a base between 0 and 1 means decay. Which choices have a base between 0 and 1?
Desmos Guide
Set a simple initial mass and enter the four models
In Desmos, define a value for the initial mass, for example M0 = 1. Then enter each option as a function of t: A(t) = M0 - 0.13t, B(t) = M0*(1.13)^t, C(t) = M0*(0.87)^t, and D(t) = M0*(0.13)^t.
Check the value after 1 hour
Create a table for t (or just type A(1), B(1), C(1), and D(1)). Since the substance loses 13% in one hour, the mass after 1 hour should be 87% of the initial mass, which is 0.87*M0 when M0 = 1. See which function gives this value at t = 1.
Confirm the decay pattern over time
Look at the graphs of the four functions for . The correct model should show a smooth exponential decrease, always staying positive and approaching 0 as gets large, with the amount lost each hour being a constant percentage, not a constant amount. Identify which graph has that behavior.
Step-by-step Explanation
Translate the percent loss into a multiplier
The substance loses 13% of its mass each hour.
- Losing 13% means it keeps of its mass each hour.
- As a decimal, .
So after each hour, the mass is multiplied by (it becomes times what it was the previous hour).
Write expressions for a few specific hours
Start with the initial mass .
- After 1 hour: .
- After 2 hours: you multiply by again, so
- After 3 hours: multiply by once more,
You can see a pattern of repeated multiplication by . The exponent matches the number of hours.
Write the general exponential model and match the option
From the pattern:
- After hours, you multiply by exactly times.
- That gives the general expression
Now compare this with the answer choices: it matches choice C, , which is the correct model for losing 13% of the mass each hour.