STEM-Master
Calculus IChapter 1: ReviewSection 1-7

Exponential Functions — Question 10

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Question 10

A certain species of insect is introduced to a new habitat. Its population is modeled by the exponential function: P(t)=500⋅e0.6tP(t) = 500 \cdot e^{0.6t} where P(t)P(t) is the population size after tt weeks.

  • (a) What will the population be after 5 weeks?

  • (b) After how many weeks will the population exceed 10,000?

  • (c) How fast is the population growing at t=5t = 5 weeks?

Round all answers to two decimal places.

Original worksheet page 1: question and worked solution for 1-7-010
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Question 10 - Solution

Use the exact exponential model and keep full precision until the final rounding.

(a)

P(5)=500e3P(5)=500e^3

10042.77insects\boxed{10042.77\quad\text{insects}}

(b)

t*=ln⁡(10000/500)/0.6t_*=\ln(10000/500)/0.6

4.99weeks\boxed{4.99\quad\text{weeks}}

(c)

P′(t)=300e0.6t,P′(5)=300e3P'(t)=300e^{0.6t},\quad P'(5)=300e^3

6025.66insects/week\boxed{6025.66\quad\text{insects/week}}

The population equals 1000010000 at t=t*t=t_* and exceeds it precisely when t>t*t>t_*. The rounded threshold is not an exact inequality boundary.

Original worksheet page 2: question and worked solution for 1-7-010

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