Question
upstudy study bank question image url

The number of bacteria \( P(h) \) in a certain population increases according to the following function, where time \( h \) is measured in hours. \[ P(h)=2000 e^{0.06 h} \] How many hours will it take for the number of bacteria to reach 3200 ? Round your answer to the nearest tenth, and do not round any intermediate computations.

Ask by Barker Frank. in the United States
Mar 14,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

It will take approximately 7.8 hours for the number of bacteria to reach 3200.

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Beyond the Answer

To find the number of hours \( h \) it takes for the bacterial population \( P(h) \) to reach 3200, we start with the equation: \[ 2000 e^{0.06 h} = 3200 \] To isolate \( e^{0.06 h} \), we divide both sides by 2000: \[ e^{0.06 h} = \frac{3200}{2000} = 1.6 \] Next, we take the natural logarithm of both sides: \[ \ln(e^{0.06 h}) = \ln(1.6) \] Using the property of logarithms that \( \ln(e^x) = x \), we get: \[ 0.06 h = \ln(1.6) \] Now we can solve for \( h \) by dividing both sides by 0.06: \[ h = \frac{\ln(1.6)}{0.06} \] Calculating \( \ln(1.6) \): \[ \ln(1.6) \approx 0.4700 \] Now substituting back into the equation for \( h \): \[ h \approx \frac{0.4700}{0.06} \approx 7.8333 \] Rounding to the nearest tenth, we find: \[ h \approx 7.8 \] Thus, it will take approximately 7.8 hours for the number of bacteria to reach 3200.

Related Questions

Latest Pre Calculus Questions

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy