Question

A radioactive substance is decaying at a rate of 5 percent per day. This means that every day it looses 5 percent of its total weight. The substance starts with a weight of 80 grams. What equation can represent the decay of the substance?(1 point)
f (x)=80(1.05)^x-1
f (x)= 80(0.995)^x-1
f (x) 80(0.95)^x-1
f (x)= 80(0.5)^x-1

Answer 1

To model the decay of a radioactive substance that loses 5% of its weight per day, we start with an initial weight of 80 grams. Since it loses 5% of its weight, it retains 95% of its weight each day.

This can be expressed mathematically as:

\[ f(x) = 80(0.95)^x \]

where:
- \( f(x) \) is the weight of the substance after \( x \) days,
- \( 80 \) is the initial weight,
- \( 0.95 \) represents the remaining weight after losing 5% each day,
- \( x \) is the number of days.

Now let's look at the options given:

1. \( f (x)=80(1.05)^{x-1} \) - incorrect, this represents growth.
2. \( f (x)= 80(0.995)^{x-1} \) - incorrect, as it represents a decay of only 0.5% not 5%.
3. \( f (x) 80(0.95)^{x-1} \) - incorrect syntax, but it closely resembles the correct decay model.
4. \( f (x)= 80(0.5)^{x-1} \) - incorrect, this represents a much stronger decay.

Since option 3 looks like it is meant to be \( f(x) = 80(0.95)^x \) (with a minor syntax error regarding the "=" sign), that would be the correct model.

However, none of the provided options formally expresses the accurate decay equation without minor mistakes. The closest correct equation should be:

\[ f(x) = 80(0.95)^x \]

If you are required to select from the choices provided, it appears that option 3 is the intent behind the correct equation, though presented incorrectly.