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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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
Jayne
4.4106 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)
When time, X = 770 ms, atoms, T = 3000
3000 = a + b * \log(770)
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 isT = a + b * \log(X) a \approx -25252.862 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.
Given:
When time, X = 700 ms, atoms, T = 1500
When time, X = 770 ms, atoms, T = 3000
Step 2: Solve equations (1) and (2) simultaneously to find "a" and "b".
Subtract equation (1) from equation (2):
Substitute the value of "b" back into equation (1):
Step 3: Substitute the values of "a" and "b" into the regression model.
The regression model is
Step 4: Find the time it takes to reach 10,000 atoms using the regression model.
Answer: It will take approximately 9.543 milliseconds to reach 10,000 atoms based on the logarithmic regression model.
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