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An element with mass 330 grams decays by \( 25.5 \% \) per minute. How much of the element is remaining after 5 minutes, to the nearest 1oth of a gram?

Ask by Tyler Williams. in the United States
Jan 24,2025

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Answer

After 5 minutes, approximately 75.7 grams of the element remain.

Solution

To find the amount of the element remaining after 5 minutes, we can use the concept of exponential decay. Given: - Initial mass of the element = 330 grams - Decay rate per minute = 25.5% We can calculate the remaining mass after 5 minutes using the formula for exponential decay: \[ \text{Remaining mass} = \text{Initial mass} \times (1 - \text{Decay rate})^{\text{Time}} \] Substitute the given values into the formula: \[ \text{Remaining mass} = 330 \times (1 - 0.255)^5 \] Now, we can calculate the remaining mass after 5 minutes. Calculate the value by following steps: - step0: Calculate: \(330\left(1-0.255\right)^{5}\) - step1: Subtract the numbers: \(330\times 0.745^{5}\) - step2: Convert the expressions: \(330\left(\frac{149}{200}\right)^{5}\) - step3: Simplify: \(330\times \frac{149^{5}}{200^{5}}\) - step4: Rewrite the expression: \(10\times 33\times \frac{149^{5}}{200^{5}}\) - step5: Rewrite the expression: \(10\times 33\times \frac{149^{5}}{100^{5}\times 2^{5}}\) - step6: Rewrite the expression: \(10\times 33\times \frac{149^{5}}{10^{10}\times 2^{5}}\) - step7: Reduce the numbers: \(33\times \frac{149^{5}}{10^{9}\times 2^{5}}\) - step8: Multiply the fractions: \(\frac{33\times 149^{5}}{32\times 10^{9}}\) The remaining mass after 5 minutes is approximately 75.73 grams. Therefore, after 5 minutes, 75.73 grams of the element will remain, to the nearest 1/10th of a gram.

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To find out how much of the element is remaining after 5 minutes, we can use the formula for exponential decay: \[ N(t) = N_0 \cdot (1 - r)^t \] where \( N_0 \) is the initial amount (330 grams), \( r \) is the decay rate (25.5\% or 0.255), and \( t \) is the time in minutes (5 minutes). Plugging in the numbers gives us: \[ N(5) = 330 \cdot (1 - 0.255)^5 \] \[ N(5) = 330 \cdot (0.745)^5 \] \[ N(5) = 330 \cdot 0.186 \] \[ N(5) \approx 61.1 \text{ grams} \] So, after 5 minutes, approximately 61.1 grams of the element is remaining when rounded to the nearest tenth of a gram. One interesting fact is that exponential decay occurs in a variety of contexts beyond just radioactive substances. For instance, similar principles are applied in factors such as the depreciation of car value over time, or even in how a population of bacteria decreases when antibiotics are administered. If you're ever curious about how decay rates manifest in real-life scenarios, think about food preservation! Various foods rot or decay at different rates, influenced by factors like temperature, moisture, and exposure to air. Monitoring these factors can help you keep your groceries fresh longer!

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