Question

Element X decays radioactively with a half life of 15 minutes. If there are 600 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 280 grams?

Answer 1

Explanation:

N(t) = N₀ * (1/2)^(t / T)

Where:

N(t) is the amount of the element at time t.

N₀ is the initial amount of the element.

t is the time elapsed.

T is the half-life of the element.

In this case, N₀ is 600 grams, and we want to find out how long it takes for N(t) to reach 280 grams. We know that the half-life (T) is 15 minutes.

280 = 600 * (1/2)^(t / 15)

Now, let's solve for t:

(1/2)^(t / 15) = 280 / 600

(1/2)^(t / 15) = 7/15

To solve for t, we can take the natural logarithm (ln) of both sides:

ln((1/2)^(t / 15)) = ln(7/15)

Using the property of logarithms, we can bring down the exponent:

(t / 15) * ln(1/2) = ln(7/15)

Now, solve for t:

t / 15 = ln(7/15) / ln(1/2)

t = 15 * (ln(7/15) / ln(1/2))

t ≈ 31.6 minutes

So, it would take approximately 31.6 minutes (to the nearest tenth of a minute) for Element X to decay from 600 grams to 280 grams.