1.8: Variation - Constructing and Solving Equations (2024)

  1. Last updated
  2. Save as PDF
  • Page ID
    59372
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

    \( \newcommand{\vectorC}[1]{\textbf{#1}}\)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}}\)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}\)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    Solving Problems involving Direct, Inverse, and Joint variation

    Certain relationships occur so frequently in applied situations that they are given special names. Variation equations show how one quantity changes in relation to other quantities. The relationship between the quantities can be described as direct, inverse, or joint variation.

    Direct Variation

    Many real-world problems encountered in the sciences involve two types of functional relationships. The first type of functional relationship can be explored using the fact that the distance \(s\) in feet an object falls from rest, without regard to air resistance, can be approximated using the following formula:

    \(s=16t^{2}\)

    Here \(t\) represents the time in seconds the object has been falling. For example, after \(2\) seconds the object will have fallen \(s = 16 ( 2 ) ^ { 2 } = 16 \cdot 4 = 64\) feet.

    Time \(t\) in seconds 0 1 2 3 4
    Distancein feet
    \(s = 16 t ^ { 2 }\)
    0 16 64 144 256

    In this example, we can see that the distance varies over time as the product of a constant \(16\) and the square of the time \(t\). This relationship is described as direct variationand \(16\) is called the constant of variation or theconstant of proportionality.

    Definition: Direct Variation (\(y=kx\))

    Direct variation is a relationship where quantities behave in a like manner. If oneincreases, so does the other. If one decreases, so does the other.

    For two quantities \(x\) and \(y\), this relationship is describedas "\(y\) varies directly as \(x\)" or "\(y\) is directly proportional to \(x\)".

    The equation that describes this relationship is \(y=kx\), where \(k\) is a non-zero constant called the constant of variation or the proportionality constant.

    1.8: Variation - Constructing and Solving Equations (1)Howto: Solve a Variation Problem.

    1. Translate the given English statement containing the words variesor proportional, into a model equation.
    2. Substitute a given set of values into the equation and solve for \(k\), the constant of variation.
    3. Rewrite the equation obtained in step 1 as a formula with a value for \(k\) found in step 2 defined. Make note of the units used for each variable in the formula.
    4. Use the equation from step 3, and another set of values (with one value missing)to solve for the unknown quantity.

    Example \(\PageIndex{1}\): Direct Variation

    An object’s weight on Earth varies directly to its weight on the Moon. If a man weighs \(180\) pounds on Earth, then he will weigh \(30\) pounds on the Moon. Set up an algebraic equation that expresses the weight on Earth in terms of the weight on the Moon and use it to determine the weight of a woman on the Moon if she weighs \(120\) pounds on Earth.

    Solution

    Step 1. Translate“the weight on Earth varies directly to the weight on the Moon.” \(E = kM \)

    Step 2. Find \(k\) using "If a man weighs \(180\) pounds on Earth, then he will weigh \(30\) pounds on the Moon." \(E=180\) pounds, \(M=30\) pounds

    \(\begin{array} { cr } E = kM & \text{Model equation} \\180=k \cdot 30\\\frac { 180 } { 30 } = k \\ { 6 = k } \end{array}\)

    Step 3. The formulais \(E = 6M \), where \(E\) is the weight on Earth in pounds and \(M\) is the weight on the moon in pounds.

    Step 4. Answer the question: "determine the weight of a woman on the Moon if she weighs \(120\) pounds on Earth."\(E=120\) pounds, find M

    \(\begin{array} { cll }E = 6M & \text{Formula:} & \text{ \(E\) pounds on Earth}\\ && \text{ \(M\) pounds on the Moon}\\{ 120 = 6 M } \\ { \frac { 120 } { 6 } = M } \\ { 20 = M } \end{array}\)

    Answer:

    The woman weighs \(20\) pounds on the Moon.

    Indirect Variation

    The second functional relationship can be explored using the model that relates the intensity of light \(I\) to the square of the distance from its source \(d\).

    \(I = \frac { k } { d ^ { 2 } }\)

    Here \(k\) represents some constant. A foot-candle is a measurement of the intensity of light. One foot-candle is defined to be equal to the amount of illumination produced by a standard candle measured one foot away. For example, a \(125\)-Watt fluorescent growing light is advertised to produce \(525\) foot-candles of illumination. This means that at a distance \(d=1\) foot, \(I=525\) foot-candles and we have:

    \(\begin{array} { l } { 525 = \frac { k } { ( 1 ) ^ { 2 } } } \\ { 525 = k } \end{array}\)

    Using \(k=525\) we can construct a formula which gives the light intensity produced by the bulb:

    \(I = \frac { 525 } { d ^ { 2 } }\)

    Distance \(t\) in feet 1 2 3 4 5
    Light Intensity \(I = \frac { 525 } { d ^ { 2 } }\) 525 131.25 58.33 32.81 21

    Here \(d\) represents the distance the growing light is from the plants. In the chart above, we can see that the amount of illumination fades quickly as the distance from the plants increases.

    This type of relationship is described as an inverse variation. We say that I is inversely proportionalto the square of the distance \(d\), where \(525\) is the constant of proportionality.

    Definition: Indirect Variation (\(y=\frac{k}{x}\))

    Indirect variation is a relationship between quantities where if oneincreases, the other decreases.

    For two quantities \(x\) and \(y\), this relationship is describedas "\(y\) varies indirectly as \(x\)" or "\(y\) is inversely proportional to \(x\)".

    The equation that describes this relationship is \(y=\dfrac{k}{x}\), where \(k\) is a non-zero constant called the constant of variation or the proportionality constant.

    Example \(\PageIndex{2}\): Indirect Variation

    The weight of an object varies inversely as the square of its distance from the center of Earth. If an object weighs \(100\) pounds on the surface of Earth (approximately \(4,000\) miles from the center), how much will it weigh at \(1,000\) miles above Earth’s surface?

    Solution

    Step 1. Translate “\(w\) varies inversely as the square of \(d\)” \(w = \frac { k } { d ^ { 2 } }\)

    Step 2. Find \(k\) using "An object weighs \(100\) pounds on the surface of Earth, approximately \(4,000\) miles from the center". \(w = 100\) when \(d = 4,000\)

    \(\begin{aligned} \color{Cerulean}{( 4,000 ) ^ { 2 }}\color{black}{ \cdot} 100 & =\color{Cerulean}{ ( 4,000 ) ^ { 2 }}\color{black}{ \cdot} \frac { k } { ( 4,000 ) ^ { 2 } } \\ 1,600,000,000 &= k \\ 1.6 \times 10 ^ { 9 } &= k \end{aligned}\)

    Step 3. The formula is \(w = \frac { 1.6 \times 10 ^ { 9 } } { d ^ { 2 } }\), where \(w\) is the weight of the object in pounds and \(d\) is the distance of the object from the center of the Earth in miles.

    Step 4. Answer the question: "how much will it weigh at \(1,000\) miles above Earth’s surface?"
    Since the object is \(1,000\) miles above the surface,the distance of the object from the center of Earth is \(d = 4,000 + 1,000 = 5,000 \:\:\text{miles}\)

    \(\begin{aligned} y & = \frac { 1.6 \times 10 ^ { 9 } } { ( \color{OliveGreen}{5,000}\color{black}{ )} ^ { 2 } } \\ & = \frac { 1.6 \times 10 ^ { 9 } } { 25,000,000 } \\ & = \frac { 1.6 \times 10 ^ { 9 } } { 2.5 \times 10 ^ { 9 } } \\ & = 0.64 \times 10 ^ { 2 } \\ & = 64 \end{aligned}\)

    Answer:

    The object will weigh \(64\) pounds at a distance \(1,000\) miles above the surface of Earth.

    Joint Variation

    Lastly, we define relationships between multiple variables.

    Definition: Joint Variation and Combined Variation

    Joint variation is a relationship in whichone quantityis proportional to the product of two or more quantities.

    Combined variation exists when combinations of direct and/or inverse variation occurs

    Example \(\PageIndex{3}\): Joint Variation

    The area of an ellipse varies jointly as \(a\), half of the ellipse’s major axis, and \(b\), half of the ellipse’s minor axis as pictured. If the area of an ellipse is \(300π cm^{2}\), where \(a=10\) cm and \(b=30\) cm, what is the constant of proportionality? Give a formula for the area of an ellipse. 1.8: Variation - Constructing and Solving Equations (2)

    Solution

    Step 1. If we let \(A\) represent the area of an ellipse, then we can use the statement “area varies jointly as \(a\) and \(b\)” to write

    \(A=kab\)

    Step 2. To find the constant of variation \(k\), use the fact that the area is \(300π\) when \(a=10\) and \(b=30\).

    \(\begin{array} { c } { 300 \pi = k ( \color{OliveGreen}{10}\color{black}{ )} (\color{OliveGreen}{ 30}\color{black}{ )} } \\ { 300 \pi = 300 k } \\ { \pi = k } \end{array}\)

    Step 3. Therefore, the formula for the area of an ellipse is

    \(A=πab\)

    Answer:

    The constant of proportionality is \(π\) and the formula for the area of an ellipse is \(A=abπ\).

    1.8: Variation - Constructing and Solving Equations (3)Try It\(\PageIndex{3}\): Combined Variation

    Given that \(y\) varies directly as the square of \(x\) and inversely with \(z\), where \(y=2\) when \(x=3\) and \(z=27\), find \(y\) when \(x=2\) and \(z=16\).

    Answer
    \(\frac{3}{2}\)
    1.8: Variation - Constructing and Solving Equations (2024)

    References

    Top Articles
    Freeman brothers remorseful 20 years after murdering their parents
    Inside The Brutal Murders Of The Freeman Family - Grunge
    Promotional Code For Spades Royale
    123 Movies Black Adam
    Professor Qwertyson
    Words From Cactusi
    Lichtsignale | Spur H0 | Sortiment | Viessmann Modelltechnik GmbH
    Music Archives | Hotel Grand Bach - Hotel GrandBach
    Needle Nose Peterbilt For Sale Craigslist
    Olivia Ponton On Pride, Her Collection With AE & Accidentally Coming Out On TikTok
    Katherine Croan Ewald
    라이키 유출
    Jellyfin Ps5
    Zack Fairhurst Snapchat
    Talbots.dayforce.com
    Weepinbell Gen 3 Learnset
    Metro Pcs.near Me
    Google Doodle Baseball 76
    X-Chromosom: Aufbau und Funktion
    Jet Ski Rental Conneaut Lake Pa
    Skip The Games Fairbanks Alaska
    ABCproxy | World-Leading Provider of Residential IP Proxies
    Violent Night Showtimes Near Century 14 Vallejo
    Military life insurance and survivor benefits | USAGov
    How Taraswrld Leaks Exposed the Dark Side of TikTok Fame
    Craftybase Coupon
    Town South Swim Club
    Uno Fall 2023 Calendar
    Desales Field Hockey Schedule
    Nurtsug
    Mrstryst
    Walter King Tut Johnson Sentenced
    Chattanooga Booking Report
    Craigslist In Myrtle Beach
    2012 Street Glide Blue Book Value
    Muma Eric Rice San Mateo
    Regis Sectional Havertys
    F9 2385
    10 Rarest and Most Valuable Milk Glass Pieces: Value Guide
    COVID-19/Coronavirus Assistance Programs | FindHelp.org
    Woody Folsom Overflow Inventory
    Quaally.shop
    Gli italiani buttano sempre più cibo, quasi 7 etti a settimana (a testa)
    Ups Customer Center Locations
    Advance Auto.parts Near Me
    Mejores páginas para ver deportes gratis y online - VidaBytes
    Horseneck Beach State Reservation Water Temperature
    French Linen krijtverf van Annie Sloan
    O'reilly's On Marbach
    Craigslist Indpls Free
    Philasd Zimbra
    Worlds Hardest Game Tyrone
    Latest Posts
    Article information

    Author: Fr. Dewey Fisher

    Last Updated:

    Views: 5869

    Rating: 4.1 / 5 (42 voted)

    Reviews: 81% of readers found this page helpful

    Author information

    Name: Fr. Dewey Fisher

    Birthday: 1993-03-26

    Address: 917 Hyun Views, Rogahnmouth, KY 91013-8827

    Phone: +5938540192553

    Job: Administration Developer

    Hobby: Embroidery, Horseback riding, Juggling, Urban exploration, Skiing, Cycling, Handball

    Introduction: My name is Fr. Dewey Fisher, I am a powerful, open, faithful, combative, spotless, faithful, fair person who loves writing and wants to share my knowledge and understanding with you.