Consider the shape below. (1) determine the value of x so that the volume of the shape is 165cm cube, and (2) once you have found the value of x, determine the surface are if the shape. be sure to show all steps and details of how you found your answer.

Step-by-step explanation:

The volume of the solid with x=0 is the product of its length, width, and height:

  V0 = (10 cm)(3 cm)(8 cm) = 240 cm³

The volume of the cutout is the same:

  Vc = (x cm)(3 cm)(x cm) = 3x² cm³

Then the volume of the solid with the cutout is ...

  V1 = V0 -Vc = 240 cm³ - 3x² cm³

This is 165 cm³, so we have ...

  165 = 240 -3x²

  55 = 80 -x² . . . . . divide by 3

  x² = 80 -55 = 25 . . . . add x²-55

  x = √25 = 5 . . . . . take the square root

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The surface area is the total of the front and back faces and the lateral area of all the faces between them. That lateral area is the product of the perimeter of the front (or back) face and the width (3 cm). The perimeter (p) is the total of all edge lengths around the front face:

  p = 10 + 8 + (10-x)/2 + x + x + x + (10-x)/2 + 8 = 36 +2x

For x=5, the perimeter is ...

  p = 36 +2·5 = 46

so the lateral area is ...

... la = (46 cm)(3 cm) = 138 cm².

The area of each of the front and back faces is the area of the overal rectangular shape (10 cm by 8 cm) less the area of the cutout (x² cm²). So, that face area is ...

  fa = 2×((10 cm)(8 cm) -(5 cm)²) = 2×(55 cm²) = 110 cm²

Then the total surface area is the sum of lateral area and face area:

  S = la + fa = 138 cm² + 110 cm² = 248 cm²

It is better to use completing the square method to solve this quadratic equation.