An element with a mass of 910 grams decays by 27.4% per minute. To the nearest 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 910 grams decays by  27.4% per minute. To the nearest minute, how long will it be until there are 10 grams of the element remaining? Answer:

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Identify amounts and rate: Identify the initial amount, final amount, and the decay rate per minute. Initial amount (a) = 910 grams Final amount (y) = 10 grams Decay rate per minute = 27.4%Convert rate to decimal: Convert the decay rate from a percentage to a decimal for calculation purposes. Decay rate per minute = 27.4% = 27.4/100 = 0.274Use exponential decay formula: Use the exponential decay formula y = a * (1 - r)^t, where y is the final amount, a is the initial amount, r is the decay rate, and t is the time in minutes. We need to solve for t when y = 10 grams, a = 910 grams, and r = 0.274. 10 = 910 * (1 - 0.274)^tIsolate exponential part: Divide both sides of the equation by 910 to isolate the exponential part of the equation. 10/910 = (1 - 0.274)^tSimplify left side: Simplify the left side of the equation. 10/910 ≈ 0.01098901 0.01098901 = (1 - 0.274)^tTake natural logarithm: Take the natural logarithm (ln) of both sides to solve for t. ln(0.01098901) = ln((1 - 0.274)^t)Bring down exponent: Use the property of logarithms that ln(a^b) = b * ln(a) to bring down the exponent t. ln(0.01098901) = t * ln(1 - 0.274)Calculate logarithms: Calculate ln(1 - 0.274) and ln(0.01098901) using a calculator. ln(1 - 0.274) ≈ ln(0.726) ≈ -0.319 ln(0.01098901) ≈ -4.512Solve for t: Divide both sides of the equation by ln(1 - 0.274) to solve for t. t = ln(0.01098901) / ln(1 - 0.274) t ≈ -4.512 / -0.319Round to nearest minute: Calculate the value of t using the values from the previous step. t ≈ -4.512 / -0.319 t ≈ 14.144Since we cannot have a fraction of a minute in this context, we round t to the nearest whole minute. t ≈ 14 minutes

An element with a mass of $910$ grams decays by $27.4 \%$ per minute. To the nearest 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 $910$ grams decays by $27.4 \%$ per minute. To the nearest minute, how long will it be until there are $10$ grams of the element remaining? $\newline$ Answer: