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Differential Equation and Decay The amount in grams of a certain radioactive material present after \(t\) years is given by the function \(P(t)\). Match each of the following answers with its corresponding question. Answers a. Solve \(P(t)=.5 P(0)\) for \(t\). b. Solve \(P(t)=.5\) for \(t\). c. \(P(.5)\) d. \(P^{\prime}(.5)\) e. \(P(0)\) f. Solve \(P^{\prime}(t)=-.5\) for \(t\). g. \(y^{\prime}=k y\) h. \(P_{0} e^{k t}, k<0\) Questions A. Give a differential equation satisfied by \(P(t)\). B. How fast will the radioactive material be disintegrating in \(\frac{1}{2}\) year? C. Give the general form of the function \(P(t)\). D. Find the half-life of the radioactive material. E. How many grams of the material will remain after \(\frac{1}{2}\) year? F. When will the radioactive material be disintegrating at the rate of \(\frac{1}{2}\) gram per year? G. When will there be \(\frac{1}{2}\) gram remaining? \(\mathbf{H}\). How much radioactive material was present initially?

Step by step solution

01

- Match Question A

Question A asks for a differential equation satisfied by \(P(t)\). The typical form of a differential equation for radioactive decay is \(y^{\textprime}=k y\). Therefore, the answer is g.

02

- Match Question B

Question B asks how fast the radioactive material will be disintegrating in \(\frac{1}{2}\) year. To find this, determine the derivative at \(t = 0.5\). This corresponds to \(P^{\textprime}(0.5)\). Therefore, the answer is d.

03

- Match Question C

Question C asks for the general form of the function \(P(t)\). The general form for an exponentially decaying function is \(P_{0} e^{k t}, k<0\). Therefore, the answer is h.

04

- Match Question D

Question D asks for the half-life of the material. To find this, solve \(P(t) = 0.5 P(0)\). Therefore, the answer is a.

05

- Match Question E

Question E asks how many grams of the material will remain after \(\frac{1}{2}\) year. This can be found by evaluating \(P(t)\) at \(t = 0.5\). Therefore, the answer is c.

06

- Match Question F

Question F asks when the material will be disintegrating at the rate of \( \frac{1}{2} \) gram per year. To find this, solve \(P^{\textprime}(t) = -0.5\). Therefore, the answer is f.

07

- Match Question G

Question G asks when there will be \( \frac{1}{2} \) gram remaining. This involves solving \(P(t) = 0.5\). Therefore, the answer is b.

08

- Match Question H

Question H asks how much radioactive material was initially present. This is given by \(P(0)\). Therefore, the answer is e.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative of Decay Functions

The derivative of a decay function \(P(t) = P_0 e^{kt}\) provides the decay rate at any given time \(t\). Using differentiation rules, the derivative is found as follows:
\[ P^{\textprime}(t) = \frac{d}{dt} (P_0 e^{kt}) = P_0 k e^{kt} \] Given \(k\) is negative, this derivative shows how quickly the material is decaying over time.
For instance, if we evaluate the derivative at specific points in time, we can determine how rapidly the material is disintegrating at those moments. This rate of change is crucial for knowing how the quantity of a radioactive substance evolves continuously.