{"id":1040,"date":"2025-07-21T22:50:34","date_gmt":"2025-07-21T22:50:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1040"},"modified":"2026-01-14T19:33:41","modified_gmt":"2026-01-14T19:33:41","slug":"variation-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/variation-learn-it-2\/","title":{"raw":"Variation: Learn It 2","rendered":"Variation: Learn It 2"},"content":{"raw":"

Inverse Variation<\/h2>\r\nInverse variation describes a relationship where one variable increases as the other decreases. This concept is crucial in understanding how different quantities affect each other inversely.\r\n\r\n
\r\n

inverse variation<\/h3>\r\nIn an inverse variation, the relationship between two variables [latex]x[\/latex] and [latex]y[\/latex] can be expressed as:\r\n

[latex]y = \\dfrac{k}{x^n}[\/latex]<\/p>\r\nwhere [latex]k[\/latex] is a nonzero constant, then we say that [latex]y[\/latex] varies inversely<\/strong> with the power of [latex]x[\/latex].\r\n[latex]\\\\[\/latex]\r\n\r\nIn inversely proportional<\/strong> relationships, or inverse variations<\/strong>, there is a constant multiple [latex]k = x^n \\cdot y[\/latex].\r\n\r\n<\/section>

Water temperature in an ocean varies inversely<\/em> to the water\u2019s depth. Between the depths of [latex]250[\/latex] feet and [latex]500[\/latex] feet, the formula [latex]T=\\frac{14,000}{d}[\/latex] gives us the temperature in degrees Fahrenheit at a depth in feet below Earth\u2019s surface.\r\n[latex]\\\\[\/latex]\r\nConsider the Atlantic Ocean, which covers [latex]22\\%[\/latex] of Earth\u2019s surface. At a certain location, at the depth of [latex]500[\/latex] feet, the temperature may be [latex]28^\\circ\\text{F}[\/latex].If we create a table\u00a0we observe that, as the depth increases, the water temperature decreases.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]d[\/latex], depth<\/th>\r\n[latex]T=\\frac{\\text{14,000}}{d}[\/latex]<\/th>\r\nInterpretation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
[latex]500[\/latex] ft<\/td>\r\n[latex]\\frac{14,000}{500}=28[\/latex]<\/td>\r\nAt a depth of [latex]500[\/latex] ft, the water temperature is [latex]28^\\circ\\text{F}[\/latex].<\/td>\r\n<\/tr>\r\n
[latex]350[\/latex] ft<\/td>\r\n[latex]\\frac{14,000}{350}=40[\/latex]<\/td>\r\nAt a depth of [latex]350[\/latex] ft, the water temperature is [latex]40^\\circ\\text{F}[\/latex].<\/td>\r\n<\/tr>\r\n
[latex]250[\/latex] ft<\/td>\r\n[latex]\\frac{14,000}{250}=56[\/latex]<\/td>\r\nAt a depth of [latex]250[\/latex] ft, the water temperature is [latex]56^\\circ\\text{F}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formula [latex]y=\\dfrac{k}{x}[\/latex] for inverse variation in this case uses [latex]k=14,000[\/latex].\r\n\r\n\"Graph\r\n\r\nWe notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be inversely proportional<\/strong> and each term varies inversely<\/strong> with the other. Inversely proportional relationships are also called inverse variations<\/strong>.\r\n\r\n<\/section>
A tourist plans to drive [latex]100[\/latex] miles. Find a formula for the time the trip will take as a function of the speed the tourist drives.[reveal-answer q=\"81111\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"81111\"]Recall that multiplying speed by time gives distance. If we let [latex]t[\/latex]\u00a0represent the drive time in hours, and [latex]v[\/latex]\u00a0represent the velocity (speed or rate) at which the tourist drives, then [latex]vt=[\/latex]\u00a0distance. Because the distance is fixed at [latex]100[\/latex] miles, [latex]vt=100[\/latex]. Solving this relationship for the time gives us our function.\r\n

[latex]\\begin{align}t\\left(v\\right)&=\\dfrac{100}{v} \\\\[1mm] &=100{v}^{-1} \\end{align}[\/latex]<\/p>\r\nWe can see that the constant of variation is [latex]100[\/latex] and, although we can write the relationship using the negative exponent, it is more common to see it written as a fraction.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

How To: Given a description of an inverse variation problem, solve for an unknown.\r\n<\/strong>\r\n
    \r\n \t
  1. Identify the input, [latex]x[\/latex], and the output, [latex]y[\/latex].<\/li>\r\n \t
  2. Determine the constant of variation. You may need to multiply [latex]y[\/latex]\u00a0by the specified power of [latex]x[\/latex]\u00a0to determine the constant of variation.<\/li>\r\n \t
  3. Use the constant of variation to write an equation for the relationship.<\/li>\r\n \t
  4. Substitute known values into the equation to find the unknown.<\/li>\r\n<\/ol>\r\n<\/section>
    A quantity [latex]y[\/latex]\u00a0varies inversely with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]\u00a0when [latex]x=2[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is [latex]6[\/latex].[reveal-answer q=\"482072\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"482072\"]The general formula for inverse variation with a cube is [latex]y=\\dfrac{k}{{x}^{3}}[\/latex]. The constant can be found by multiplying [latex]y[\/latex]\u00a0by the cube of [latex]x[\/latex].\r\n

    [latex]\\begin{align}k&={x}^{3}y \\\\[1mm] &={2}^{3}\\cdot 25 \\\\[1mm] &=200 \\end{align}[\/latex]<\/p>\r\nNow we use the constant to write an equation that represents this relationship.\r\n

    [latex]\\begin{align}y&=\\dfrac{k}{{x}^{3}},\\hspace{2mm}k=200 \\\\[1mm] y&=\\dfrac{200}{{x}^{3}} \\end{align}[\/latex]<\/p>\r\nSubstitute [latex]x=6[\/latex]\u00a0and solve for [latex]y[\/latex].\r\n

    [latex]\\begin{align}y&=\\dfrac{200}{{6}^{3}} \\\\[1mm] &=\\dfrac{25}{27} \\end{align}[\/latex]<\/p>\r\nAnalysis of the Solution<\/strong>\r\n\r\nThe graph of this equation is a rational function.\r\n\r\n\"Graph\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]318959[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]318960[\/ohm_question]<\/section>","rendered":"

    Inverse Variation<\/h2>\n

    Inverse variation describes a relationship where one variable increases as the other decreases. This concept is crucial in understanding how different quantities affect each other inversely.<\/p>\n

    \n

    inverse variation<\/h3>\n

    In an inverse variation, the relationship between two variables [latex]x[\/latex] and [latex]y[\/latex] can be expressed as:<\/p>\n

    [latex]y = \\dfrac{k}{x^n}[\/latex]<\/p>\n

    where [latex]k[\/latex] is a nonzero constant, then we say that [latex]y[\/latex] varies inversely<\/strong> with the power of [latex]x[\/latex].
    \n[latex]\\\\[\/latex]<\/p>\n

    In inversely proportional<\/strong> relationships, or inverse variations<\/strong>, there is a constant multiple [latex]k = x^n \\cdot y[\/latex].<\/p>\n<\/section>\n

    Water temperature in an ocean varies inversely<\/em> to the water\u2019s depth. Between the depths of [latex]250[\/latex] feet and [latex]500[\/latex] feet, the formula [latex]T=\\frac{14,000}{d}[\/latex] gives us the temperature in degrees Fahrenheit at a depth in feet below Earth\u2019s surface.
    \n[latex]\\\\[\/latex]
    \nConsider the Atlantic Ocean, which covers [latex]22\\%[\/latex] of Earth\u2019s surface. At a certain location, at the depth of [latex]500[\/latex] feet, the temperature may be [latex]28^\\circ\\text{F}[\/latex].If we create a table\u00a0we observe that, as the depth increases, the water temperature decreases.<\/p>\n\n\n\n\n\n\n\n
    [latex]d[\/latex], depth<\/th>\n[latex]T=\\frac{\\text{14,000}}{d}[\/latex]<\/th>\nInterpretation<\/th>\n<\/tr>\n<\/thead>\n
    [latex]500[\/latex] ft<\/td>\n[latex]\\frac{14,000}{500}=28[\/latex]<\/td>\nAt a depth of [latex]500[\/latex] ft, the water temperature is [latex]28^\\circ\\text{F}[\/latex].<\/td>\n<\/tr>\n
    [latex]350[\/latex] ft<\/td>\n[latex]\\frac{14,000}{350}=40[\/latex]<\/td>\nAt a depth of [latex]350[\/latex] ft, the water temperature is [latex]40^\\circ\\text{F}[\/latex].<\/td>\n<\/tr>\n
    [latex]250[\/latex] ft<\/td>\n[latex]\\frac{14,000}{250}=56[\/latex]<\/td>\nAt a depth of [latex]250[\/latex] ft, the water temperature is [latex]56^\\circ\\text{F}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    We say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formula [latex]y=\\dfrac{k}{x}[\/latex] for inverse variation in this case uses [latex]k=14,000[\/latex].<\/p>\n

    \"Graph<\/p>\n

    We notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be inversely proportional<\/strong> and each term varies inversely<\/strong> with the other. Inversely proportional relationships are also called inverse variations<\/strong>.<\/p>\n<\/section>\n

    A tourist plans to drive [latex]100[\/latex] miles. Find a formula for the time the trip will take as a function of the speed the tourist drives.<\/p>\n