Question

an element with mass 970 grams decays by 27.7% per minute. how much of the element is remaining after 13 minutes, to the nearest 10th of a gram?

Explanation:

Step1: Identify the decay formula

The formula for exponential decay is \( A = P(1 - r)^t \), where \( A \) is the final amount, \( P \) is the initial amount, \( r \) is the rate of decay (as a decimal), and \( t \) is the time.

Step2: Convert the decay rate to a decimal

The decay rate is \( 27.7\% \), so \( r = 0.277 \).

Step3: Substitute the values into the formula

We have \( P = 970 \), \( r = 0.277 \), and \( t = 13 \). Plugging these into the formula gives \( A = 970(1 - 0.277)^{13} \).

Step4: Calculate \( (1 - 0.277) \)

\( 1 - 0.277 = 0.723 \).

Step5: Calculate \( 0.723^{13} \)

Using a calculator, \( 0.723^{13} \approx 0.0134 \) (this is an approximation, more precise calculation is needed).

Step6: Multiply by the initial amount

\( A = 970 \times 0.0134 \approx 13.0 \) (wait, actually, more accurately, let's do the calculation properly. Let's calculate \( 0.723^{13} \) more accurately. Using a calculator, \( 0.723^{13} \approx 0.013402 \). Then \( 970 \times 0.013402 \approx 13.0 \)? Wait, no, let's do it step by step. Let's compute \( 0.723^1 = 0.723 \), \( 0.723^2 = 0.723 \times 0.723 = 0.522729 \), \( 0.723^3 = 0.522729 \times 0.723 \approx 0.377933 \), \( 0.723^4 = 0.377933 \times 0.723 \approx 0.273246 \), \( 0.723^5 = 0.273246 \times 0.723 \approx 0.197557 \), \( 0.723^6 = 0.197557 \times 0.723 \approx 0.142834 \), \( 0.723^7 = 0.142834 \times 0.723 \approx 0.103269 \), \( 0.723^8 = 0.103269 \times 0.723 \approx 0.074663 \), \( 0.723^9 = 0.074663 \times 0.723 \approx 0.053981 \), \( 0.723^{10} = 0.053981 \times 0.723 \approx 0.03903 \), \( 0.723^{11} = 0.03903 \times 0.723 \approx 0.02822 \), \( 0.723^{12} = 0.02822 \times 0.723 \approx 0.02040 \), \( 0.723^{13} = 0.02040 \times 0.723 \approx 0.01475 \). Then \( 970 \times 0.01475 \approx 14.3075 \). Wait, maybe my initial approximation was wrong. Let's use a calculator for \( 0.723^{13} \). Using a calculator, \( 0.723^{13} \approx 0.01475 \). Then \( 970 \times 0.01475 \approx 14.3 \). Wait, let's use a calculator for the exact value. Let's compute \( 0.723^{13} \):

Using a calculator, \( 0.723^{13} \approx e^{13 \ln(0.723)} \). \( \ln(0.723) \approx -0.324 \), so \( 13 \times (-0.324) = -4.212 \), \( e^{-4.212} \approx 0.015 \). Then \( 970 \times 0.015 \approx 14.55 \). Wait, maybe I should use a calculator directly. Let's use a calculator to compute \( 970*(0.723)^13 \). Let's do this:

First, \( 0.723^13 \):

Using a calculator, \( 0.723^{13} \approx 0.0147 \) (more accurately, let's use a calculator: 0.723^13 = e^(13*ln(0.723)) ≈ e^(13*(-0.3240)) ≈ e^(-4.212) ≈ 0.015). Then 970*0.015 = 14.55. Wait, maybe my step-by-step multiplication was wrong. Let's use a calculator for the exact value. Let's compute 0.723^13:

0.723^2 = 0.522729

0.723^3 = 0.522729 * 0.723 ≈ 0.377933

0.723^4 = 0.377933 * 0.723 ≈ 0.273246

0.723^5 = 0.273246 * 0.723 ≈ 0.197557

0.723^6 = 0.197557 * 0.723 ≈ 0.142834

0.723^7 = 0.142834 * 0.723 ≈ 0.103269

0.723^8 = 0.103269 * 0.723 ≈ 0.074663

0.723^9 = 0.074663 * 0.723 ≈ 0.053981

0.723^10 = 0.053981 * 0.723 ≈ 0.03903

0.723^11 = 0.03903 * 0.723 ≈ 0.02822

0.723^12 = 0.02822 * 0.723 ≈ 0.02040

0.723^13 = 0.02040 * 0.723 ≈ 0.01475

Then 970 * 0.01475 = 14.3075, which rounds to 14.3 when rounded to the nearest tenth. Wait, but let's check with a calculator. Let's use a calculator to compute 970*(0.723)^13. Let's do this:

First, calculate 0.723^13:

Using a calculator, 0.723^13 ≈ 0.014702

Then 970 * 0.014702 ≈ 14.26094, which rounds to 14.3 when rounded to the nearest tenth.

Wait, maybe I made a mistake earlier. Let's confirm with a calculator. Let's use a calculator to compute 0.723^13:

0.723^1 = 0.723

0.723^2 = 0.723 * 0.723 = 0.522729

0.723^3 = 0.522729 * 0.723 = 0.377933067

0.723^4 = 0.377933067 * 0.723 = 0.273245607

0.723^5 = 0.273245607 * 0.723 = 0.197556574

0.723^6 = 0.197556574 * 0.723 = 0.142833403

0.723^7 = 0.142833403 * 0.723 = 0.103268550

0.723^8 = 0.103268550 * 0.723 = 0.074663162

0.723^9 = 0.074663162 * 0.723 = 0.053981466

0.723^10 = 0.053981466 * 0.723 = 0.039030600

0.723^11 = 0.039030600 * 0.723 = 0.028229124

0.723^12 = 0.028229124 * 0.723 = 0.020409657

0.723^13 = 0.020409657 * 0.723 = 0.014756182

Then 970 * 0.014756182 = 970 * 0.014756182 ≈ 14.31349654, which rounds to 14.3 when rounded to the nearest tenth.

Answer:

\( \boxed{14.3} \)