Question

A certain radioactive isotope decays at a rate of 2% per 100 years. if t represents time in years and y represents the amount of the isotope left then the equation for the situation is y=y0e-0.0002t . in how many years will there be 89% of the isotope left? round to the nearest year.

Answer 1

The time required for 89% of a certain radioactive isotope to remain, based on its decay rate, can be calculated using the provided decay equation, and the result approximately is 583 years when rounded to the nearest year.

Step-by-step explanation:

The situation described involves the exponential decay of a radioactive isotope, represented by the decay equation y = y0e-0.0002t, where y is the amount remaining, y0 is the initial amount, and t is time in years. We want to determine the time t when 89% of the isotope remains. To do this, we set y to 0.89y0 and solve for t.

Starting with the equation:
0.89y0 = y0e-0.0002t,

we can divide both sides by y0 to get:
0.89 = e-0.0002t.

Taking the natural logarithm of both sides gives ln(0.89) = -0.0002t.

Now we can solve for t by dividing both sides by -0.0002 to get t = ln(0.89) / -0.0002.

t = 583

After calculating the value of t, we find that the time required for 89% of the isotope to remain is approximately 583 years, when rounded to the nearest year.

Answer 2

We rewrite the equation:
y = y0 * e ^ (- 0.0002 * t)
In this equation:
I = represents the initial amount of the isotope
We have then:
89% of the isotope left:
0.89 * y0 = y0 * e ^ (- 0.0002 * t)
We clear the time:
e ^ (- 0.0002 * t) = 0.89
Ln (e ^ (- 0.0002 * t)) = Ln (0.89)
-0.0002 * t = Ln (0.89)
t = Ln (0.89) / (- 0.0002)
t = 582.6690813
round to the nearest year:
t = 583 years
Answer:
There will be 89% of the isotope left in about:
t = 583 years