text: # Differential calculus question
  
  1. Jul 8, 2014
  
  ### noahsdev
  
  1. The problem statement, all variables and given/known data
  The number of termites in a colony is increasing at a rate proportional to the number present on any day. If the number of termites increases by 25% in 100 days, how much longer (to the nearest day) will it be until the population is double the initial number?
  
  2. The attempt at a solution
  $\frac{dT}{dt}$=at
  T=∫at
  =$\frac{at^{2}}{2}$ + c
  T(x)=T
  T(0)*$\frac{5}{4}$ = T(100)
  => a=$\frac{c}{20000}$
  
  I think I am approaching this wrong. Help is appreciated,
  thanks.
  
  **UPDATE: I overlooked a something, I think I figured it out now. I got t=200, does that seem right?
  
  Last edited: Jul 8, 2014
  2. Jul 8, 2014
  
  ### ManiFresh
  
  Your initial equation seems to be wrong. The rate of change of the population is proportional to the population (on the same day).
  
  3. Jul 8, 2014
  
  ### Ray Vickson
  
  Your expression $T(t) = a t^2/2 + c$ is incorrect; it does NOT satisfy the condition that the derivative $dT/dt$ is proportional to $T$ itself.
  
  4. Jul 8, 2014
  
  ### HallsofIvy
  
  Staff Emeritus
  ManiFresh and Ray Vickson's point is that this is not the correct equation. Saying that "the rate is proportional to the number present" is $\frac{dT}{dt}= aT$.
  The "T" on the right is the number of termites, not the time in days.
  
  5. Jul 8, 2014
  
  ### noahsdev
  
  So how do I solve it? I mean, are you saying the rate of change is dT/dt = aT?
  
  6. Jul 8, 2014
  
  ### Ray Vickson
  
  I cannot see how to answer this question without doing your problem for you. I suggest you look through your textbook or course notes to find an answer. You could also do a Google search under "growth models", for example.
  
  7. Jul 8, 2014
  
  ### Staff: Mentor
  
  What is the derivative of ln(y) with respect to t, if y is a function of t?
  
  Chet
  
  8. Jul 8, 2014
  
  ### HallsofIvy
  
  Staff Emeritus
  You need to get both "dT" and "T" on the same side of the equation, "dt" on the other side.
explanation: The document is a forum post discussing a differential calculus problem. It contains a question about the rate of increase of a termite population and an attempt at a solution. The user also provides an update with a proposed answer. The post includes feedback from other users pointing out an error in the initial approach.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Differential calculus question
  
  1. Jul 8, 2014
  
  ### noahsdev
  
  1. The problem statement, all variables and given/known data
  The number of termites in a colony is increasing at a rate proportional to the number present on any day. If the number of termites increases by 25% in 100 days, how much longer (to the nearest day) will it be until the population is double the initial number?
  
  2. The attempt at a solution
  $\frac{dT}{dt}$=at
  T=∫at
  =$\frac{at^{2}}{2}$ + c
  T(x)=T
  T(0)*$\frac{5}{4}$ = T(100)
  => a=$\frac{c}{20000}$
  
  I think I am approaching this wrong. Help is appreciated,
  thanks.
  
  **UPDATE: I overlooked a something, I think I figured it out now. I got t=200, does that seem right?
  
  Last edited: Jul 8, 2014
  2. Jul 8, 2014
  
  ### ManiFresh
  
  Your initial equation seems to be wrong. The rate of change of the population is proportional to the population (on the same day).
  
  3. Jul 8, 2014
  
  ### Ray Vickson
  
  Your expression $T(t) = a t^2/2 + c$ is incorrect; it does NOT satisfy the condition that the derivative $dT/dt$ is proportional to $T$ itself.
  
  4. Jul 8, 2014
  
  ### HallsofIvy
  
  Staff Emeritus
  ManiFresh and Ray Vickson's point is that this is not the correct equation. Saying that "the rate is proportional to the number present" is $\frac{dT}{dt}= aT$.
  The "T" on the right is the number of termites, not the time in days.
  
  5. Jul 8, 2014
  
  ### noahsdev
  
  So how do I solve it? I mean, are you saying the rate of change is dT/dt = aT?
  
  6. Jul 8, 2014
  
  ### Ray Vickson
  
  I cannot see how to answer this question without doing your problem for you. I suggest you look through your textbook or course notes to find an answer. You could also do a Google search under "growth models", for example.
  
  7. Jul 8, 2014
  
  ### Staff: Mentor
  
  What is the derivative of ln(y) with respect to t, if y is a function of t?
  
  Chet
  
  8. Jul 8, 2014
  
  ### HallsofIvy
  
  Staff Emeritus
  You need to get both "dT" and "T" on the same side of the equation, "dt" on the other side.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a forum post discussing a differential calculus problem. It contains a question about the rate of increase of a termite population and an attempt at a solution. The user also provides an update with a proposed answer. The post includes feedback from other users pointing out an error in the initial approach.
  
  Label: question-and-answer