Use a linear approximation (or differentials) to estimate the given number. (Round your answer to five decimal places.) [tex] \sqrt[3

Question

Use a linear approximation (or differentials) to estimate the given number. (Round your answer to five decimal places.)
\sqrt[3 ]{28}
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Ladonna 5 years 2021-07-15T09:57:21+00:00 2 Answers 764 views 0

Answers ( )

  1. diemkieu
    0
    2021-07-15T09:58:25+00:00
    Reply

    Answer:

    Mix fraction: 3 1/27

    Improper fraction: 82/27

    Decimal approximation: 3.037

    Step-by-step explanation:

    What value is close to 28 that is a perfect cube…27 which equals 3^3.

    So let’s find the tangent line to the curve y=cubert(x) at x=27.

    We will use this equation to approximate what happens at x=28.

    First let’s rewrite the radical in our equation;

    y=x^(1/3)

    Now differentiate

    y’=(1/3) x^(1/3-1) by power rule

    Simplify

    y’=(1/3) x^(-2/3) or (1)/(3x^[2/3])

    So the slope of our tangent line at x=27 is (1)/(3(27)^[2/3])=1/(3(3)^2)=1/(3×9)=1/27.

    We will also need a point on this tangent line….We know we have the point at x=27 because that is what our tangent line to curve is being found at.

    So at x=27, we have y=cubert(27)=3. We used our equation y=cubert(x) here.

    So we want to find the equation of the line that contains point (27,3) and has slope 1/27.

    Point-slope form is

    y-y1=m(x-x1)

    Plug in our values

    y-3=(1/27)(x-27)

    Add 3 on both sides

    y=3+(1/27)(x-27)

    We will use this linear equation to approximate cubert(28) by replacing x with 28.

    y=3+(1/27)(28-27)

    y=3+(1/27)(1)

    y=3+1/27

    You can write that as a mix fraction if you want.

    This value is than 3 but super close to 3 since 1/27 is close to 0.

    Mix fraction: 3 1/27

    Improper fraction: 82/27

    Decimal approximation: 3.037

    Cubert of 28 when smashed into calculator as is gives approximately 3.0366 which is pretty close to our approximation.

  2. Xavia
    -1
    2021-07-15T09:58:37+00:00
    Reply

    the answer is either one of these all are correct according to what i got

    3 1/27

    82/27

    3.037

    hope dis helped

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