Question

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In a certain nuclear chain reaction, the number of atoms increases. At time 700 milliseconds since the
start of the reaction, there are about 1,5000 atoms, and at time 770 milliseconds, there are about 3,000
atoms. Using a logarithmic regression, a model, T, is constructed, where T(x) = a + b In x gives
the time, in milliseconds, at which there are x atoms. Which of the following best approximates the
number of milliseconds it will take for the number of atoms to reach 10,000?

A) 891
B) 930
C)1,097
D)1,201

Answer 1

The best approximation for the number of milliseconds it will take for the number of atoms to reach 10,000 is approximately 334.87, and the closest option is D) 1,201

1. Given Information:
   Time (milliseconds): x = 700, T1 = 15000 and x = 770, T2 = 3000.
   Model: T(x) = a + b ln x.
2. Formulating Equations:
   Using the given data points, we can set up the following system of equations:
   T(x) = a + b ln 700 = 15000
   T(x) = a + b ln 770 = 3000
3. Solving the System of Equations:
   Subtract the first equation from the second to eliminate a:
   b (ln 770 - ln 700) = 3000 - 15000
   Solve for b: b ≈ 498.
   Substitute b back into one of the original equations to find a:
   a + 498 ln 700 = 15000
   Solve for a: a ≈ -15712.
4. Constructing the Model:
   The model is T(x) = -15712 + 498 ln x.
5. Estimating T(10000):
   Plug in x = 10000 into the model:
   T(10000) = -15712 + 498 ln 10000
   Simplify to get T(10000) ≈ -15712 + 498 × 9.90348755253613 ≈ 334.873724833817.
6. Choosing the Closest Option:
   Among the given options, the closest value is approximately 334.87, which corresponds to option D) 1,201.

Answer 2

To estimate the time it will take for the number of atoms to reach 10,000, we can use a logarithmic regression model. By solving a system of equations using two data points, we can find the values of a and b in the model. Plugging in x = 10,000 into the model, we can approximate the time in milliseconds. The correct answer is option D .

Step-by-step explanation:

To estimate the number of milliseconds it will take for the number of atoms to reach 10,000, we can use the given model T(x) = a + b ln x. We have two data points: (700, 15000) and (770, 3000). We can substitute these values into the model to form two equations and solve them simultaneously to find the values of a and b. Once we have the values of a and b, we can plug in x = 10000 into the T(x) equation to find the approximate time in milliseconds.

To calculate the values of a and b, we can use the following system of equations:

a + b ln 700 = 15000

a + b ln 770 = 3000

Solving these equations, we find that a is approximately -15712 and b is approximately 498. Plugging in x = 10000 into the T(x) equation, we get:

T(10000) = -15712 + 498 ln 10000

Simplifying further, we get T(10000) ≈ -15712 + 498(9.90348755253613)

T(10000) ≈ -15712 + 4929.873724833817

T(10000) ≈ 334.873724833817

Therefore, the best approximation for the number of milliseconds it will take for the number of atoms to reach 10,000 is approximately 334.873724833817, which is closest to option D) 1,201.