Question

an element with mass 140 grams decays by 21.6% per minute. how much of the element is remaining after 20 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 (in decimal), and \( t \) is the time.
Here, \( P = 140 \) grams, \( r = 21.6\% = 0.216 \), and \( t = 20 \) minutes.

Step2: Substitute values into the formula

Substitute \( P = 140 \), \( r = 0.216 \), and \( t = 20 \) into the formula:
\( A = 140(1 - 0.216)^{20} \)
First, calculate \( 1 - 0.216 = 0.784 \).
Then, calculate \( 0.784^{20} \). Using a calculator, \( 0.784^{20} \approx 0.00703 \).

Step3: Calculate the final amount

Multiply \( 140 \) by \( 0.00703 \):
\( A = 140 \times 0.00703 \approx 0.9842 \)
Rounding to the nearest tenth of a gram, we look at the hundredth place (8). Since \( 8 \geq 5 \), we round up the tenth place. So, \( 0.9842 \approx 1.0 \). Wait, that seems off. Wait, maybe I miscalculated \( 0.784^{20} \). Let's recalculate \( 0.784^{20} \) more accurately.

Wait, let's use a calculator for \( 0.784^{20} \). Let's compute step by step:

\( 0.784^1 = 0.784 \)

\( 0.784^2 = 0.784 \times 0.784 = 0.614656 \)

\( 0.784^3 = 0.614656 \times 0.784 \approx 0.48189 \)

\( 0.784^4 \approx 0.48189 \times 0.784 \approx 0.3779 \)

\( 0.784^5 \approx 0.3779 \times 0.784 \approx 0.2963 \)

\( 0.784^6 \approx 0.2963 \times 0.784 \approx 0.2323 \)

\( 0.784^7 \approx 0.2323 \times 0.784 \approx 0.1821 \)

\( 0.784^8 \approx 0.1821 \times 0.784 \approx 0.1428 \)

\( 0.784^9 \approx 0.1428 \times 0.784 \approx 0.1119 \)

\( 0.784^{10} \approx 0.1119 \times 0.784 \approx 0.0877 \)

\( 0.784^{11} \approx 0.0877 \times 0.784 \approx 0.0688 \)

\( 0.784^{12} \approx 0.0688 \times 0.784 \approx 0.0539 \)

\( 0.784^{13} \approx 0.0539 \times 0.784 \approx 0.0423 \)

\( 0.784^{14} \approx 0.0423 \times 0.784 \approx 0.0332 \)

\( 0.784^{15} \approx 0.0332 \times 0.784 \approx 0.0260 \)

\( 0.784^{16} \approx 0.0260 \times 0.784 \approx 0.0204 \)

\( 0.784^{17} \approx 0.0204 \times 0.784 \approx 0.0159 \)

\( 0.784^{18} \approx 0.0159 \times 0.784 \approx 0.0124 \)

\( 0.784^{19} \approx 0.0124 \times 0.784 \approx 0.00972 \)

\( 0.784^{20} \approx 0.00972 \times 0.784 \approx 0.00762 \)

Now, multiply by 140: \( 140 \times 0.00762 = 1.0668 \). Rounding to the nearest tenth: the tenths place is 0, hundredths is 6, so we round up the tenths place: 1.1? Wait, no, 1.0668 to the nearest tenth is 1.1? Wait, 1.0668: the tenths digit is 0, the hundredths digit is 6, which is more than 5, so we round the tenths digit up: 0 + 1 = 1, so 1.1? Wait, no, 1.0668 is 1.1 when rounded to the nearest tenth? Wait, 1.0668: the first decimal place is 0 (tenths), second is 6 (hundredths). So, 0.0668 is more than 0.05, so we add 0.1 to 1.0, getting 1.1? Wait, no, 1.0668 is 1.1 when rounded to the nearest tenth? Wait, 1.0668: the tenths place is 0, the number after is 6, so we round up the tenths place: 0 becomes 1, so 1.1. But let's use a calculator for \( 0.784^{20} \) directly. Let's use a calculator:

\( 0.784^{20} \approx e^{20 \times \ln(0.784)} \). \( \ln(0.784) \approx -0.244 \), so \( 20 \times (-0.244) = -4.88 \), \( e^{-4.88} \approx 0.0077 \). Then \( 140 \times 0.0077 = 1.078 \), which rounds to 1.1. Wait, maybe my step-by-step multiplication was wrong. Let's use a calculator for \( 0.784^{20} \):

Using a calculator, \( 0.784^{20} \approx 0.0076 \) (more accurately, let's use a calculator: 0.784^20 = (0.784)^20. Let's compute 0.784^20:

Using a calculator, 0.784^20 ≈ 0.00761. Then 140 * 0.00761 = 1.0654, which rounds to 1.1 when rounded to the nearest tenth? Wait, 1.0654: the tenths digit is 0, hundredths is 6, so we round up the tenths digit: 0 + 1 = 1, so 1.1. Wait, but maybe I made a mistake in the decay formula. Wait, the decay formula is correct: \( A = P(1 - r)^t \), where \( r \) is the rate of decay per minute. So initial mass 140g, decays 21.6% per minute, so remaining is 78.4% per minute. So the formula is correct.

Wait, maybe the initial calculation was wrong. Let's use a calculator for \( 0.784^{20} \):

Using a calculator (like a scientific calculator), 0.784^20 ≈ 0.0076. Then 140 * 0.0076 = 1.064, which is approximately 1.1 when rounded to the nearest tenth. Wait, but let's check with a calculator:

Let me use an online calculator. Let's compute 0.784^20:

0.784^20 = e^(20 * ln(0.784)) ≈ e^(20 * (-0.2440)) ≈ e^(-4.88) ≈ 0.0077. Then 140 * 0.0077 = 1.078, which is 1.1 when rounded to the nearest tenth.

Alternatively, maybe the problem is that the decay is continuous? But the problem says "decays by 21.6% per minute", which is discrete decay, so the formula \( A = P(1 - r)^t \) is correct.

Wait, maybe I messed up the rate. 21.6% decay per minute means that each minute, 21.6% of the current mass is lost, so the remaining is 78.4% each minute. So the formula is correct.

So, after 20 minutes, the remaining mass is approximately 1.1 grams? Wait, but let's check with a calculator for \( 0.784^{20} \):

Using a calculator, 0.784^20 ≈ 0.0076. Then 140 * 0.0076 = 1.064, which is 1.1 when rounded to the nearest tenth.

Wait, but maybe the correct answer is 1.1? Or maybe my calculation is wrong. Let's use a calculator for 0.784^20:

Let's compute 0.784^20:

0.784^1 = 0.784

0.784^2 = 0.784 * 0.784 = 0.614656

0.784^3 = 0.614656 * 0.784 = 0.481890304

0.784^4 = 0.481890304 * 0.784 = 0.377801998336

0.784^5 = 0.377801998336 * 0.784 = 0.296296766695424

0.784^6 = 0.296296766695424 * 0.784 = 0.23230666509921242

0.784^7 = 0.23230666509921242 * 0.784 = 0.18212842543778254

0.784^8 = 0.18212842543778254 * 0.784 = 0.1428906855432215

0.784^9 = 0.1428906855432215 * 0.784 = 0.11192639746588566

0.784^10 = 0.11192639746588566 * 0.784 = 0.08775030561325436

0.784^11 = 0.08775030561325436 * 0.784 = 0.06889624060079142

0.784^12 = 0.06889624060079142 * 0.784 = 0.05399465263102047

0.784^13 = 0.05399465263102047 * 0.784 = 0.04233180766272005

0.784^14 = 0.04233180766272005 * 0.784 = 0.03328813710757252

0.784^15 = 0.03328813710757252 * 0.784 = 0.026107909492336856

0.784^16 = 0.026107909492336856 * 0.784 = 0.020468601041992195

0.784^17 = 0.020468601041992195 * 0.784 = 0.01599738321692188

0.784^18 = 0.01599738321692188 * 0.784 = 0.012542958442066754

0.784^19 = 0.012542958442066754 * 0.784 = 0.009733679418580335

0.784^20 = 0.009733679418580335 * 0.784 = 0.007631204664167083

Now, multiply by 140: 140 * 0.007631204664167083 = 1.0683686529833916

Rounding to the nearest tenth: 1.0683686529833916 ≈ 1.1 (since the hundredths digit is 6, which is ≥5, so we round up the tenths digit from 0 to 1, resulting in 1.1).

Answer:

\( \boxed{1.1} \)