Question

an element with mass 310 grams decays by 8.9% per minute. how much of the element is remaining after 19 minutes, to the nearest tenth of a gram?

Explanation:

Step1: Identify the decay formula

The formula for exponential decay is \( N(t) = N_0 \times (1 - r)^t \), where \( N_0 \) is the initial amount, \( r \) is the decay rate per time unit, and \( t \) is the time. Here, \( N_0 = 310 \) grams, \( r = 0.08 \) (since 8% = 0.08), and \( t = 19 \) minutes.

Step2: Substitute the values into the formula

Substitute \( N_0 = 310 \), \( r = 0.08 \), and \( t = 19 \) into the formula:
\( N(19) = 310 \times (1 - 0.08)^{19} \)
First, calculate \( 1 - 0.08 = 0.92 \). Then, calculate \( 0.92^{19} \). Using a calculator, \( 0.92^{19} \approx 0.2018 \).

Step3: Calculate the final amount

Multiply 310 by 0.2018:
\( N(19) = 310 \times 0.2018 \approx 62.56 \) (rounded to two decimal places)

Answer:

Approximately 62.56 grams of the element remain after 19 minutes.