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Exponential Decay: Radioactive Substance and Time

Created by @nathanedwards

at November 1st 2023, 10:58:24 am.

AP_Calculus_AB-Questions

#exponential-decay

#radioactive-substance

#decay-constant

Question:

A radioactive substance decays exponentially over time. The initial amount of the substance is 500 grams, and after 4 hours, there are only 300 grams left.

a) Find the decay constant, k, for this substance.

b) Write an equation representing the amount of the substance remaining, A(t), as a function of time t.

c) Determine how much of the substance will remain after 10 hours.

Answer:

a) To find the decay constant, k, we can use the formula for exponential decay:

A(t) = A_0 \cdot e^{-kt}

Where:

A(t) is the amount of the substance remaining at time t
A₀ is the initial amount of the substance
k is the decay constant
e is Euler's number (approximately 2.71828)

We are given that A₀ = 500 grams and A(4) = 300 grams. Substituting these values into the equation, we have:

300 = 500 \cdot e^{-k \cdot 4}

To solve for k, we can divide both sides of the equation by 500 and take the natural logarithm of both sides:

\ln(0.6) = -4k

Dividing both sides of the equation by -4, we get:

k \approx \frac{\ln(0.6)}{-4}

Using a calculator, we find that k ≈ 0.1155 (rounded to four decimal places).

b) Now that we have the decay constant k, we can write the equation representing the amount of the substance remaining, A(t), as a function of time t:

A(t) = 500 \cdot e^{-0.1155t}

c) To determine how much of the substance will remain after 10 hours, we can substitute t = 10 into the equation we obtained in part b):

A(10) = 500 \cdot e^{-0.1155 \cdot 10}

Using a calculator, we find that A(10) ≈ 199.52 grams (rounded to two decimal places).

Therefore, after 10 hours, approximately 199.52 grams of the substance will remain.