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A quantity with an initial value of 8900 grows exponentially at a rate of \( 65 \% \) every hour. What is the value of the quantity after 411 minutes, to the nearest hundredth?

Ask by Lambert Allan. in the United States
Feb 25,2025

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Answer

The quantity grows to approximately 274,887.52 after 411 minutes.

Solution

To find the value of the quantity after 411 minutes, we need to first convert the time from minutes to hours since the growth rate is given in terms of hours. Given: - Initial value of the quantity = 8900 - Growth rate = 65% per hour Let's calculate the value of the quantity after 411 minutes: 1. Convert 411 minutes to hours: \[ 411 \text{ minutes} = \frac{411}{60} \text{ hours} \] 2. Calculate the value of the quantity after 411 minutes using the exponential growth formula: \[ A = P(1 + r)^t \] where: - \( A \) is the final value of the quantity - \( P \) is the initial value of the quantity - \( r \) is the growth rate - \( t \) is the time in hours Substitute the values into the formula and calculate the final value of the quantity. Calculate the value by following steps: - step0: Calculate: \(8900\left(1+0.65\right)^{\frac{411}{60}}\) - step1: Add the numbers: \(8900\times 1.65^{\frac{411}{60}}\) - step2: Reduce the fraction: \(8900\times 1.65^{\frac{137}{20}}\) - step3: Convert the expressions: \(8900\left(\frac{33}{20}\right)^{\frac{137}{20}}\) - step4: Simplify: \(8900\times \frac{33^{\frac{137}{20}}}{20^{\frac{137}{20}}}\) - step5: Multiply: \(\frac{8900\times 33^{\frac{137}{20}}}{20^{\frac{137}{20}}}\) - step6: Simplify: \(\frac{8900\times 33^{6}\sqrt[20]{33^{17}}}{20^{\frac{137}{20}}}\) - step7: Simplify: \(\frac{8900\times 33^{6}\sqrt[20]{33^{17}}}{20^{6}\sqrt[20]{20^{17}}}\) - step8: Reduce the fraction: \(\frac{89\times 33^{6}\sqrt[20]{33^{17}}}{800^{2}\sqrt[20]{20^{17}}}\) - step9: Multiply by the Conjugate: \(\frac{89\times 33^{6}\sqrt[20]{33^{17}}\times \sqrt[20]{20^{3}}}{800^{2}\sqrt[20]{20^{17}}\times \sqrt[20]{20^{3}}}\) - step10: Simplify: \(\frac{89\times 33^{6}\sqrt[20]{33^{17}}\times \sqrt[20]{8000}}{800^{2}\sqrt[20]{20^{17}}\times \sqrt[20]{20^{3}}}\) - step11: Multiply the numbers: \(\frac{89\times 33^{6}\sqrt[20]{33^{17}\times 8000}}{800^{2}\sqrt[20]{20^{17}}\times \sqrt[20]{20^{3}}}\) - step12: Multiply the numbers: \(\frac{89\times 33^{6}\sqrt[20]{33^{17}\times 8000}}{20\times 800^{2}}\) The value of the quantity after 411 minutes is approximately 274,887.52 to the nearest hundredth.

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The Deep Dive

To calculate the value of the quantity after 411 minutes of exponential growth at a rate of 65% per hour, we first convert the time into hours. Since there are 60 minutes in an hour, 411 minutes is equivalent to \( \frac{411}{60} \approx 6.85 \) hours. We can use the exponential growth formula: \[ A = P(1 + r)^t \] where: - \( A \) is the amount after time \( t \), - \( P \) is the initial amount (8900), - \( r \) is the rate of growth (0.65), - \( t \) is the time in hours (6.85). Plugging in the values: \[ A = 8900(1 + 0.65)^{6.85} \] This simplifies to: \[ A = 8900(1.65)^{6.85} \] Calculating \( (1.65)^{6.85} \) gives approximately 31.967. Now, we multiply this result by 8900: \[ A \approx 8900 \times 31.967 \approx 284,740.30 \] Therefore, the value of the quantity after 411 minutes is approximately **$284,740.30**. --- Remember, exponential growth can lead to some eye-popping numbers, so buckle up – especially at 65% per hour! In the real world, you can see exponential growth in things like investments, populations in ideal conditions, and even the spread of information online, where what starts small can rapidly balloon out of control! And speaking of control, keep an eye on your calculations! It’s easy to mix up your inputs, especially with rates and time conversions. Double-checking your figures can prevent some “oops” moments in math and finance, leading to a smoother journey through the numerical landscape!

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