For each problem, create a function to model the scenario.
Solution
Here’sLet how’s to analyze model and each of the scenarios create in functions for each the scenario provided in given the image problems.:
** Problem4 4.
A homeThe’s home value’s value \( V V ) is is $200,000 \$ and200 increases, by000, increasing by 1.1.5% per year t:
The5 function is exponential growth%, modeled as:
per year t: **
The function is an exponential growth model:
[
\$$
cdSimplotified (:
1\ +[
V0.015)^(t)t = 200,
000 \ \cd]
Simplotified (:
1.015 \)^t[
\ V =]
200,---
000### \ **5cd. A populationot \( ( p1 \.)015 of)^t
\ fruit]
---
### Problem 5
The population $$p$$ of fruit flies starts at 150 flies and is currently at grows150 by and increases205% by 205% every every day day $$\( d d \$$:):
**
-A A 205205%% increase increase means is the equivalent new to population multiplying is the \( current amount1 by + \( 21. +05 =2 .305. =05 \3). times05 the \ previous). day The:
model\ is[
:
p(d ) \ =[
150 p \ =cd ot150 ( \3cd.ot05 ()^3d.
05\)^]
d---
### ** \6]
.---
Mr###. Problem 6 Brust’s yard is decreasing
Mr. Brust’s yard starts in with grass area by 28. 1% per1800 year square $$t$$: feet of grass \( g \**
The area decreases exponentially, where the), remaining decreasing percentage by is $$10028\.% - 28.1\% =1 %71 per. year9 \(\ t% \$$:
)- or A \( decrease of0 .28719. \1):
%\ leaves[
\(g (t100)\ =% 1800 \cdot (0 -. 719)^t28
.1\% = \71]
.---
9###\ **%7 \.), The or 0.719 of the value $$V original$$ of a motorcycle amount purchased. for The $ function11 is,:
000 decreases \ by[
15 g. = 18007 \%cd perot year ( $$0 t.$$:719**
)^The motorcycle's value decreasest, leaving
\( 100 \]
---
### Problem\ 7
The value $$V% - \15). of7\ a motorcycle starts% = 84.3\%$$ or $$0.843$$:
$$V(t) = 11,000 \cdot (0. at \(843 \$)^11t,
000\$$
)---
andWould decreases you by like me15 to. further7 elaborate% on per how year these \( formulas t are \ derived):
or- solved A? decrease
of---
###15 **.7% leaves $$100\% - 15.7\Exp% = 84.3\%$$, or anding0 Questions:**
1..843 How do you of interpret exponential the growth original vs. amount decay in. these The problems function?
is2:
. What $$V = 11,000 \cd wouldot happen ( to0 the. equations if the843 percentage rates)^t
were$$
---
changed?
3.Let me know How would if these you models’d change if the like time further periods explanations \( or t clarification \!)
were---
measuredHere in are months **five related questions instead** of to years deepen?
4. Can you rewrite these functions to solve for \( t \ your) understanding:
1 when a. How specific value is reached ( woulde the.g., $$function V change = in 300 Problem ,000$$)?
5. What4 if the growth would rate be was the compounded graphical monthly representation instead of of these yearly?
2. How would functions you over interpret time \(?
t---
=### **0Tip \:)**
inWhen these dealing scenarios with?
percentages3,. always What convert happens them to the values in Problems 6 and 7 as $$t \to \infty$$?
4. to decimal form Can ( youe rewrite.g the., exponential growth15 equations% ( =e 0.15) to simplify calculations in exponential models..g., Problems 4 and 5) as logarithmic equations to solve for $$t$$?
5. What would happen if a problem combined growth and decay in different intervals of time?
**Tip:** When modeling with exponential functions, always double-check the base—growth uses $$1 + r$$, and decay uses $$1 - r$$.
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Math Problem Analysis
Mathematical Concepts
Exponential Growth and Decay Algebraic Modeling
Formulas
Exponential Growth: V = V₀(1 + r)^t Exponential Decay: V = V₀(1 - r)^t