Find the half-life (in hours) of a radioactive substance that is reduced by 30 percent in 80 hours

amount = a(1/2)^(t/k), where a is the starting amount, and k is the half-life period

now, when t=0, amount = 1
1 = a(1/2)^0 , so a = 1
after 80 hours:
.7 = (.5)^(80/k)
log .7 = (80/k)log .5
80/k = log.7/log.5
k/80 = log.5/log.7
k = 80log.5/log.7 = about 155.5 hours

To find the half-life of a radioactive substance that is reduced by 30 percent in 80 hours, we need to use the concept of exponential decay.

The formula for exponential decay is given by:

A(t) = A₀ * (1/2)^(t/h)

Where:
A(t) is the amount of the substance remaining at time t.
A₀ is the initial amount of the substance.
t is the time that has passed since the start.
h is the half-life of the substance.

In this case, we know that the substance is reduced by 30 percent, which means that only 70 percent (100 percent - 30 percent) remains after the 80 hours.

So, we have:

0.7 = 1 * (1/2)^(80/h)

Simplifying the equation, we can rewrite it as:

0.7 = (1/2)^(80/h)

And taking the logarithm of both sides, we have:

log(0.7) = log((1/2)^(80/h))

Using logarithmic properties, the equation becomes:

log(0.7) = (80/h) * log(1/2)

To find the value of h (the half-life), we need to rearrange the equation:

h = (80 * log(1/2)) / log(0.7)

Simplifying this equation with the given values, we find:

h ≈ 231.034

Therefore, the half-life of the radioactive substance is approximately 231.034 hours.

To find the half-life of a radioactive substance, we need to determine the time it takes for the substance to reduce by half. In this case, we are given that the substance is reduced by 30 percent in 80 hours.

To calculate the half-life, follow these steps:

Step 1: Calculate the remaining amount after the given time period.
- Since the substance is reduced by 30 percent, this means 70 percent (100% - 30%) of the substance remains after 80 hours.
- Convert 70 percent to decimal form: 70% = 0.70.
- Multiply the initial amount by 0.70 to get the remaining amount after 80 hours.

Step 2: Calculate the half-life.
- To find the half-life, we need to determine the time it takes for the substance to reduce by half.
- Divide the given time period (80 hours) by the natural logarithm of 0.5 (ln(0.5)).

Let's calculate the half-life:

Step 1:
Initial amount remaining = 100% - 30% = 70% = 0.70
Initial amount remaining after 80 hours = Initial amount * Initial amount remaining = Initial amount * 0.70

Step 2:
Half-life = Given time period (hours) / ln(0.5)
Half-life = 80 hours / ln(0.5)

Now, calculate ln(0.5) ≈ -0.6931:

Half-life = 80 hours / -0.6931 ≈ -115.53 hours

The half-life of the radioactive substance is approximately 115.53 hours.