Power & Root Functions

  1. Figure out which value(s) of \(x\) satisfy each of the following algebraic equations. Express those values as decimals rounded to three figures.

    \( x^{1.23} = 17\)
    \( x \approx 10.008 \)
    \( x^{\frac{1}{5}} = 1.2345\)
    \(x \approx 2.867\)
    \( \sqrt{x+3}=-2 \)
    \(x=1 \)
    \( \sqrt[4]{x^{5}}-3 = 7\)
    \(x \approx 6.310 \)
    \( 7\sqrt[3]{2x-1} = 50\)
    \( x \approx 182.716\)
    \( 1 = 2.8(1+3x)^{4.14} \)
    \(x \approx -0.073\)
  2. Solve the following equation for \(c.\)
    \(\displaystyle x = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \)
  3. For an investment of \(P\) dollars growing at an annual interest rate of \(r\) that is compounded monthly, this following formula relates the value \(S\) of the investment after \(t\) years. \[ S = P\left(1 + \frac{r}{12}\right)^{12t} \]
    1. Solve the formula for \(r\).
    2. If you have $10,000 and would like to invest it to earn $1,475.22 in interest after two years, what annual interest rate should you look for?
  4. The wind chill index is a measurement of how cold it feels outside due to the effects of the wind whipping away surface heat from your body. During cold weather, this is the “feels like” temperature that meteorologist list in weather reports. According to the National Weather Service, the following formula relates the wind chill index \(W\) with the wind velocity \(v\) (mph) and ambient air temperature \(t\) (°F). \[ W = 35.74 + 0.6215t - (35.75 - 0.4275t)v^{0.16} \]
    1. For a fixed temperature \(t\) you can write the windchill as a function of the wind velocity, \(W(v)\). Write down a formula for \(W(v)\) for an ambient air temperature of 45°F.
    2. Using your formula for \(W(v),\) calculate how cold it feels like due to windchill on a 45°F day when the wind is gusting up to 40mph.
    3. On a 45°F day, how fast does the wind need to be blowing to make it feels like it’s 32°F?
    4. Look up the current temperature and max wind speed for today to calculate how cold it “feels like” outside due to wind chill.
  5. In the production of a plumbus, there are two main raw materials that are required: absidian and borron. It turns out that the number of units of absidian required to produce the plumbus varies directly with the cube of the amount of borron (in pounds) that is required.
    1. Define two variables, one to represent the amount of absidian and another to represent the amount of borron. Then, recalling what it means for two quantities to vary directly, write down the equation our situation suggests.
    2. Suppose \(500\) units of absidian and 5lbs of borron are required to produce 100 plumbuses. Use this information to find out what the direct variation coefficient (usually denoted as the variable \(k\)) must be.
    3. Now that we know \(k\), we know everything. If we have 10lbs of borron lying around the plumbus factory, how many units of absidian do we need to go forage to turn it all into plumbuses?