Question

Solve Steps 1 through 3 2. Graph this function for 0 ≤ t ≤ 100. Choose the graph.

Answer 1

To find the remaining amount in grams of the radioactive isotope after 25 years, let us replace the "t" in the function with 25.

`A(25)=300e^(-0.02813(25))`

Then, solve.

a. Multiply the exponents of the natural number e. The equation will become:

`A(25)=300e^(-0.70325)`

b. Apply the exponent on the natural number e. The equation will become:

`A(25)=300(0.494974)`

c. Multiply the two numbers 300 and 0.494974. The result is:

`A(25)=148.4922\approx148.49g`

Therefore, the amount of radioactive isotope remaining after 25 years is approximately 148.49 grams.

For the graph, since this is decaying, we know already that as the years go by, the amount of grams remaining decreases. Therefore, the graph should have a negative slope because we have an inverse relationship between time and the amount remaining.

Out of 4 graphs, Option B and Option D show this inverse relationship.

However, we know that the initial amount is 300 grams. This means at 0 years, the amount of the radioactive isotope should be 300 grams.

Between Options B and D, Option D shows the correct initial value which is 300 (3 dashes up). Hence, the graph of this function is best shown in Graph D.

To calculate the half-life, let's calculate first how many is half of the initial amount.

`300*(1)/(2)=150g`

Half the initial amount is 150 grams. Let's take a look at the graph how many years will it take for the isotope to decay to 150 grams.

As we can see in the graph, in about 24.64 years, the radioactive isotope will be half of its initial amount. (Option C)