Math Problem Statement

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 tt:

  • The5 function is exponential growth%, modeled as: per year tt: ** 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

Theorems

Exponential Growth and Decay Formula

Suitable Grade Level

Grades 8-10