A radioactive isotope has a decay constant of 9.68E-03 /s. Calculate the time taken for three quarters of a sample of the isotope to decay.

Answer 1

The decay constant, denoted by λ, is given as 9.68E-03 /s.

The decay of a radioactive sample can be modeled using the equation: N(t) = N0 * e^(-λt),
where N(t) is the number of radioactive atoms remaining at time t, N0 is the initial number of radioactive atoms, λ is the decay constant, and e is the base of the natural logarithm.

We need to calculate the time taken for three-quarters (3/4) of the sample to decay, which means we want to find t when N(t) = 1/4 * N0.

1/4 * N0 = N0 * e^(-λt).

Dividing both sides of the equation by N0, we get:

1/4 = e^(-λt).

To isolate t, take the natural logarithm (ln) of both sides:

ln(1/4) = ln(e^(-λt)).

Using the property of logarithms, ln(a^b) = b * ln(a), we have:

ln(1/4) = -λt * ln(e).

Since ln(e) is equal to 1, we can simplify the equation further:

ln(1/4) = -λt.

Now, we solve for t:

-λt = ln(1/4).

Dividing both sides by -λ, we get:

t = ln(1/4) / (-λ).

Plugging in the given decay constant, λ = 9.68E-03 /s, we can now calculate t:

t = ln(1/4) / (-9.68E-03 /s).

Using a calculator, evaluate ln(1/4) / (-9.68E-03) to get the time taken for three-quarters of the sample to decay.

Answer 2

list of prime numbers from 50 through 75

Answer 3

The prime numbers from 50 through 75 are:

53, 59, 61, 67, 71, 73