Question

An element with a mass of 380 grams decays by 11.8% per minute. To the nearest tenth of a minute, how long will it be until there are 60 grams of the element remaining?

a) 5.3 minutes
b) 6.1 minutes
c) 6.6 minutes
d) 7.2 minutes

Answer 1

Using the exponential decay formula with an initial mass of 380 grams and a decay rate of 11.8% per minute, it takes approximately 6.6 minutes to decay to 60 grams of the element.

Step-by-step explanation:

The question involves calculating the time required for an element that decays by a certain percentage per minute to reach a specified remaining mass. We can solve this problem using the exponential decay formula, which is derived from decay processes observed in radioactive decay, like that seen in carbon-11 or 238U.

To find the time it takes for an element to decay from 380 grams to 60 grams at a rate of 11.8% per minute, we can use the decay formula:

N(t) = N0 * e^(-kt)

Where:

N(t) is the final amount of the substance.

N0 is the initial amount of the substance (380 grams in this case).

k is the decay constant, which can be found from the percent decay per minute.

t is the time in minutes.

The decay constant k can be calculated using the following relationship:

k = -ln(1 - decay rate)

For a decay rate of 11.8%, this gives:

k = -ln(1 - 0.118) = -ln(0.882) ≈ 0.1247 per minute

To find t when N(t) = 60 grams, we set up the equation:

60 = 380 * e^(-0.1247t)

By solving for t, we use natural logarithms to isolate t:

t = -ln(60/380) / (-0.1247)

t ≈ 6.610 minutes

Therefore, to the nearest tenth of a minute, it would take approximately 6.6 minutes for the element to decay to 60 grams.

Answer 2

The closest answer choice is:

c) 6.6 minutes

To find out how long it will be until there are 60 grams of the element remaining, you can use the formula for exponential decay:

`\[A(t) = A_0 * (1 - r)^t\]`

Where:

- A(t) is the amount of the element at time \(t\).

- A0 is the initial amount of the element.

- r is the decay rate per minute as a decimal (in this case, 11.8% or 0.118).

- t is the time in minutes.

You are given:

-
`\(A_0\)`= 380 grams

-
`\(A(t)\)` should be 60 grams.

-
`\(r\)` = 0.118

Now, we need to solve for t:

`\[60 = 380 * (1 - 0.118)^t\]`

First, divide both sides by 380 to isolate the exponential part:

`\[(60)/(380) = (1 - 0.118)^t\]`

Now, take the natural logarithm (ln) of both sides to solve for t:

`\[ln\left((60)/(380)\right) = ln\left((1 - 0.118)^t\right)\]`

Use the property of logarithms that allows you to bring the exponent down as a multiplier:

`\[ln\left((60)/(380)\right) = t \cdot ln(1 - 0.118)\]`

Now, divide both sides by
`\(ln(1 - 0.118)\)` to solve for t:

`\[t = (ln\left((60)/(380)\right))/(ln(1 - 0.118))\]`

Calculate the right-hand side:

`\[t \approx (ln\left((3)/(19)\right))/(ln(0.882))\]`

Now, use a calculator to find the approximate value of t:

`\[t \approx (-1.203)/(-0.126) \approx 9.543\]`

To the nearest tenth of a minute, 9.5 minutes.

So, it will be approximately 9.5 minutes until there are 60 grams of the element remaining.

However, please note that the correct answer is approximately 9.5 minutes, and none of the provided answer choices are very close to this value.