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Chapter 16: Problem 8

A germ population has a growth curve of \(A e^{\rho .4 t}\). At what value of \(t\) does its original value double? a) \(9.682\) b) \(7.733\) c) \(4.672\) d) \(1.733\) Trigonometry

Short Answer

Expert verified
Question: A germ growth experiment showed that the population of germs follows the growth curve \(A e^{\rho .4 t}\). If the germ population doubles at some time \(t\), what is the approximate value of \(t\) given the following options: a) 9.682 b) 7.733 c) 4.672 d) 1.733 Answer: Assuming \(\rho\) equals 1, the approximate value of \(t\) would be d) 1.733. If \(\rho\) is not provided or does not equal 1, it is impossible to find a precise value for \(t\) based on the provided information.

Step by step solution

01

Understand the problem and identify the given values

Given the growth curve \(A e^{\rho .4 t} = 2A\), hence the target is to find the value of \(t\) when the germ population doubles from the original population. Here, \(A\) is the original population of germs, \(\rho\) is the growth rate and \(t\) is the time.
02

Setup the equation for \(t\)

We need to find the value of \(t\) when the growth of germs doubles the original population. This leads to the equation: \(A e^{\rho .4 t} = 2A\). From this equation, we can clearly cancel out \(A\) on both sides. Hence, the equation reduces to: \(e^{\rho .4 t} = 2\).
03

Solve the equation

Solve the equation \(e^{\rho .4 t} = 2\) for \(t\). Apply the natural logarithm ln on both sides, we have: ln(\(e^{\rho .4 t}\)) = ln(2) This simplifies to \(\rho .4 t\) = ln(2) (since ln and e are inverse operations, the left side simplifies to \(\rho .4 t\)). Solving for \(t\), we get \(t = \frac{ln(2)}{\rho .4}\).
04

Check the answer options

Using the option values provided in the problem for \(t\), where \(t\) can be either \(9.682\), or \(7.733\), or \(4.672\), or \(1.733\). Plug each of these into the equation \(t = \frac{ln(2)}{\rho .4}\) and calculate \(t\). The value that matches the closest is the correct answer. In this problem, since \(\rho\) is not provided, we are unable to directly use the equation to find \(t\).
05

Note

Noticing that the problem does not provide the value of \(\rho\), which is necessary to compute \(t\), the question must be incomplete or it assumes that \(\rho\) equals 1. If so, then the equation would be \(t = \frac{ln(2)}{.4}\), which can be further simplified or calculated, and the result can be compared with the answer options to find the closest match. If \(\rho\) equals 1, then \(t = \frac{ln(2)}{.4} \approx 1.733\) and hence option d) \(t=1.733\) would be the correct answer. If \(\rho\) does not equal 1, it's impossible to find a precise value for \(t\) based on the provided information.

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