An element with a mass of 220 grams decays by 26.6% per minute. To the nearest tenth of a minute, how long will it be until there are 10 grams of the element remaining? Answer:

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An element with a mass of 220 grams decays by  26.6% per minute. To the nearest tenth of a minute, how long will it be until there are 10 grams of the element remaining? Answer:

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Understand the decay process: Understand the decay process. The decay process can be described by the exponential decay formula: N(t) = N_0 * (1 - r)^t, where N(t) is the remaining amount after time t, N_0 is the initial amount, r is the decay rate per time period, and t is the time period.Convert to decimal: Convert the percentage decay rate to a decimal. 26.6% as a decimal is 0.266.Set up equation: Set up the equation with the given values. We have N_0 = 220 grams, N(t) = 10 grams, and r = 0.266. Plugging these into the decay formula gives us 10 = 220 * (1 - 0.266)^t.Simplify the equation: Simplify the equation. Divide both sides by 220 to isolate the exponential expression: 10/220 = (1 - 0.266)^t.Calculate left side: Calculate the left side of the equation. 10/220 simplifies to 1/22.Take natural logarithm: Take the natural logarithm of both sides to solve for t. ln(1/22) = ln((1 - 0.266)^t).Use logarithm property: Use the property of logarithms to bring down the exponent. ln(1/22) = t * ln(1 - 0.266).Calculate ln values: Calculate ln(1 - 0.266) and ln(1/22). ln(1 - 0.266) ≈ ln(0.734) and ln(1/22) ≈ ln(0.04545).Divide to solve: Divide to solve for t. t = ln(0.04545) / ln(0.734).Calculate t value: Calculate the value of t. t ≈ ln(0.04545) / ln(0.734) ≈ -3.09104 / -0.30952 ≈ 9.987, which rounds to 10.0 to the nearest tenth of a minute.

An element with a mass of $220$ grams decays by $26.6 \%$ per minute. To the nearest tenth of a minute, how long will it be until there are $10$ grams of the element remaining? $\newline$ Answer:

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An element with a mass of $220$ grams decays by $26.6 \%$ per minute. To the nearest tenth of a minute, how long will it be until there are $10$ grams of the element remaining? $\newline$ Answer: