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A gemcutter wants to trim a gemstone that is approximately a regular octahedron with sides measured in millimeters (mm). She will remove imperfections by reducing the length of each side by xx mm. The approximate new surface area is given in mm2\text{mm}^2 by the function: \newlineS(x)=3.4(56.25−15x+x2)S(x)=3.4(56.25-15x+x^2) \newlineWhat is the current length of each side?

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Q. A gemcutter wants to trim a gemstone that is approximately a regular octahedron with sides measured in millimeters (mm). She will remove imperfections by reducing the length of each side by xx mm. The approximate new surface area is given in mm2\text{mm}^2 by the function: \newlineS(x)=3.4(56.25−15x+x2)S(x)=3.4(56.25-15x+x^2) \newlineWhat is the current length of each side?
  1. Identify Function: Identify the original function for the surface area of the gemstone.\newlineThe function given is S(x)=3.4(56.25−15x+x2)S(x) = 3.4(56.25 - 15x + x^2). This function represents the new surface area after xx mm is trimmed from each side of the gemstone. To find the current length of each side, we need to consider the value of xx that has not been trimmed yet, which means we need to find the value when x=0x = 0.
  2. Substitute x=0x=0: Substitute x=0x = 0 into the function to find the surface area of the gemstone before trimming.\newlineS(0)=3.4(56.25−15(0)+02)S(0) = 3.4(56.25 - 15(0) + 0^2)\newlineS(0)=3.4(56.25)S(0) = 3.4(56.25)
  3. Calculate Original Area: Calculate the surface area of the original gemstone.\newlineS(0)=3.4×56.25S(0) = 3.4 \times 56.25\newlineS(0)=191.25 mm2S(0) = 191.25 \, \text{mm}^2
  4. Surface Area Formula: Recognize that the surface area of a regular octahedron is given by the formula S=23a2S = 2 \sqrt{3} a^2, where aa is the length of a side. We need to equate this to the original surface area we found to solve for aa.191.25=23a2191.25 = 2 \sqrt{3} a^2
  5. Isolate a2a^2: Divide both sides of the equation by 2×32 \times \sqrt{3} to isolate a2a^2.\newlinea2=191.252×3a^2 = \frac{191.25}{2 \times \sqrt{3}}
  6. Calculate a2a^2: Calculate the value of a2a^2.
    a2=191.252×3a^2 = \frac{191.25}{2 \times \sqrt{3}}
    a2=191.252×1.732a^2 = \frac{191.25}{2 \times 1.732}
    a2=191.253.464a^2 = \frac{191.25}{3.464}
    a2=55.21a^2 = 55.21 (rounded to two decimal places)
  7. Solve for aa: Take the square root of both sides to solve for aa.a=55.21a = \sqrt{55.21}a≈7.43 mma \approx 7.43 \, \text{mm}

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