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determine how many milliseconds it will take to reach 10 000 atoms based on the logarithmic regression model t a b log x g

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determine how many milliseconds it will take to reach 10,000 atoms based on the logarithmic regression model T = a + b * log(X), Given the data points: - At time 700 ms, there are about 1500 atoms - At time 770 ms, there are 3000 atoms

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Answer to a math question determine how many milliseconds it will take to reach 10,000 atoms based on the logarithmic regression model T = a + b * log(X), Given the data points: - At time 700 ms, there are about 1500 atoms - At time 770 ms, there are 3000 atoms

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106 Answers

Step 1: Find the values of "a" and "b" using the given data points.

Given:
When time, X = 700 ms, atoms, T = 1500

1500 = a + b * \log(700)

...... (1)

When time, X = 770 ms, atoms, T = 3000

3000 = a + b * \log(770)

...... (2)

Step 2: Solve equations (1) and (2) simultaneously to find "a" and "b".

Subtract equation (1) from equation (2):

3000 - 1500 = b * (\log(770) - \log(700))

1500 = b * \log\left(\frac{770}{700}\right)

1500 = b * \log\left(\frac{77}{70}\right)

1500 = b * \log(1.1)

1500 = b * 0.0414

b = \frac{1500}{0.0414}

b \approx 36232.92

Substitute the value of "b" back into equation (1):

1500 = a + 36232.92 * \log(700)

a = 1500 - 36232.92 * \log(700)

a \approx -25252.862

Step 3: Substitute the values of "a" and "b" into the regression model.

The regression model is

T = a + b * \log(X)

where

a \approx -25252.862

and

b \approx 36232.92

.

Step 4: Find the time it takes to reach 10,000 atoms using the regression model.

10000 = -25252.862 + 36232.92 * \log(X)

36232.92 * \log(X) = 10000 + 25252.862

36232.92 * \log(X) = 35252.862

\log(X) = \frac{35252.862}{36232.92}

\log(X) \approx 0.9725

X = 10^{\log(X)}

X \approx 10^0.9725

X \approx 9.543

Answer: It will take approximately 9.543 milliseconds to reach 10,000 atoms based on the logarithmic regression model.

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