Question

An element with a mass of 370 grams decays by 29.3% per minute. To the nearest tenth of a minute, how long will it be until there are 2 grams of the element remaining?

Answer 1

It will take approximately (15.1) minutes for the element to decay to (2) grams from its initial mass of (370) grams.

To solve this problem, we'll use the concept of exponential decay, which is described by the formula:

`\[ P(t) = P_0 * e^((-kt)) \]`

Where:

- P(t) is the amount of substance remaining after time t ,

-
`\( P_0 \)` is the initial amount of the substance,

- e is the base of the natural logarithm,

- k is the decay constant,

- t is time.

However, we're given the decay rate as a percentage per minute, not in the form of a decay constant. Let's convert this to a decay constant first.

Step 1: Find the Decay Constant ( k )

The decay rate per minute is 29.3%, which means 70.7% of the substance remains after each minute. The decay constant \( k \) can be found using the relationship:

`\[ P(t) = P_0 * (1 - \text{decay rate})^t \]`

After 1 minute,
`\( P(t) = P_0 * 0.707 \).`We set t = 1 and solve for k using the exponential decay formula:

`\[ P_0 * e^((-k)) = P_0 * 0.707 \]`

Step 2: Calculate Time (t ) When 2 grams Remain

We need to find t when P(t) = 2 grams. The initial amount
`\( P_0 \)` is 370 grams. Using the exponential decay formula:

`\[ 2 = 370 * e^((-kt)) \]`

We'll solve for t in this equation.

Let's begin by calculating the decay constant k .

The decay constant (k ) for the element is approximately 0.3467 per minute.

Step 3: Solve for Time (t) When 2 Grams Remain

Now, we'll use the exponential decay formula to solve for t when ( P(t) = 2 ) grams:

`\[ 2 = 370 * e^((-0.3467t)) \]`

We rearrange and solve for t :

`\[ e^((-0.3467t)) = (2)/(370) \]`

`\[ -0.3467t = \ln\left((2)/(370)\right) \]`

`\[ t = -(\ln\left((2)/(370)\right))/(0.3467) \]`

Let's calculate ( t ).

It will take approximately (15.1) minutes (to the nearest tenth of a minute) for the element to decay to (2) grams from its initial mass of (370) grams.