Question

The function M(t)=170⋅5⁻ᵗ gives the mass in kilograms of a radioactive substance after t years.

Write the function M in the form M(t)=M₀e⁻ᵏᵗ and give the values of M₀ and k.
M₀ =____kilograms, k=____years⁻¹

Answer 1

The function M(t)=170⋅5−t can be rewritten as M(t)=M₀e−kt by finding the appropriate decay constant k. By taking the natural logarithm of 5, we find that k ≈ 1.60944 years−1 and M₀ is 170 kilograms.

Step-by-step explanation:

The given function for the mass of a radioactive substance is M(t)=170⋅5−t. To write this function in the form of M(t)=M₀e−kt, we need to find the values of M₀ and k that will make the two expressions equivalent.

Firstly, we observe that the base of the natural logarithm, e, is approximately 2.71828182... Hence, to express the given function using e as the base, we need to find a value for k such that 5−t is equivalent to e−kt.

Since 5 is e raised to some power, we can find this power using the natural logarithm of 5. This gives us the value of k:

k = ln(5) ≈ 1.60944

So the equation becomes M(t)=170e−(1.60944)t, thus M₀ is 170 kilograms and k is approximately 1.60944 years−1.

Note: To find the exact value of k for specific isotopes, we need the decay constant (λ), which is related to the half-life (T₁/2) of the isotope.

The relationship between k and the half-life is given by k = ln(2) / T₁/2, where ln(2) is the natural logarithm of 2, equivalent to 0.693.