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Estimate the value of 44.08 / (79.06 - 19.54) to 2 significant figures and calculate the percentage error.

Mathematics

ScanSolve AI Answer

Step-by-step solution

8 steps
  1. Round each number to 2 significant figures.

    • 44.08≈4444.08 \approx 44
    • 79.06≈7979.06 \approx 79
    • 19.54≈2019.54 \approx 20
  2. Evaluate the estimated value.

    Estimated Value=4479−20=4459\text{Estimated Value} = \frac{44}{79 - 20} = \frac{44}{59} Estimated Value≈0.74576\text{Estimated Value} \approx 0.74576
  3. Calculate the actual value of the expression.

    Actual Value=44.0879.06−19.54=44.0859.52\text{Actual Value} = \frac{44.08}{79.06 - 19.54} = \frac{44.08}{59.52} Actual Value≈0.74059\text{Actual Value} \approx 0.74059
  4. Determine the percentage error.

    The formula for percentage error is ∣Actual Value−Estimated ValueActual Value∣×100%\left| \frac{\text{Actual Value} - \text{Estimated Value}}{\text{Actual Value}} \right| \times 100\%.

    Percentage Error=∣0.74059−0.745760.74059∣×100%\text{Percentage Error} = \left| \frac{0.74059 - 0.74576}{0.74059} \right| \times 100\% Percentage Error=∣−0.005170.74059∣×100%\text{Percentage Error} = \left| \frac{-0.00517}{0.74059} \right| \times 100\% Percentage Error=0.006981×100%\text{Percentage Error} = 0.006981 \times 100\% Percentage Error≈0.698%\text{Percentage Error} \approx 0.698\%

    The estimated value is 0.74576\boxed{0.74576} and the percentage error is 0.698%\boxed{0.698\%}.

  5. Rearrange the equation of the circle by grouping xx terms and yy terms, and moving the constant to the right side.

    The given equation is x2+y2+6x−8y−2=0x^2 + y^2 + 6x - 8y - 2 = 0.

    (x2+6x)+(y2−8y)=2(x^2 + 6x) + (y^2 - 8y) = 2
  6. Complete the square for the xx terms and yy terms.

    For x2+6xx^2 + 6x, add (62)2=32=9\left(\frac{6}{2}\right)^2 = 3^2 = 9.
    For y2−8yy^2 - 8y, add (−82)2=(−4)2=16\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16.
    Add these values to both sides of the equation.

    (x2+6x+9)+(y2−8y+16)=2+9+16(x^2 + 6x + 9) + (y^2 - 8y + 16) = 2 + 9 + 16
  7. Rewrite the expressions as squared terms.

    (x+3)2+(y−4)2=27(x + 3)^2 + (y - 4)^2 = 27
  8. Identify the coordinates of the center and the radius.

    The standard form of a circle's equation is (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2, where (h,k)(h,k) is the center and rr is the radius.
    Comparing with (x+3)2+(y−4)2=27(x + 3)^2 + (y - 4)^2 = 27:
    The center (h,k)(h,k) is (−3,4)(-3, 4).
    The radius squared r2=27r^2 = 27, so the radius r=27=9×3=33r = \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}.

    The coordinates of the center are (−3,4)\boxed{(-3, 4)} and the radius is 33\boxed{3\sqrt{3}}.

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