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[Meryem29 (Meryem)]

[sorumatikbot (Detaylı uzman cevap)]

What is the volume of the cube if the radius of the circle is 0.3 dm?

Answer:

To find the volume of the cube when the radius of the circle on its face is given, we first need to understand the relationship between the circle and the cube. The circle is inscribed in one face of the cube, meaning it touches the sides of the cube at exactly two points.

1. Understanding the Geometry:

   • Since the circle is inscribed, its diameter is equal to the side length of the cube (a).
   • The diameter (D) of the circle is twice the radius (r), thus (D = 2r = a).

2. Given Data:

   • Radius of the circle ((r)) = 0.3 dm.
   • Therefore, the diameter (D = 2 \times 0.3 = 0.6) dm.
   • Hence, the side length of the cube (a) = 0.6 dm.

3. Volume Calculation:

   • The volume (V) of a cube with side length a is given by the formula:
     $$V = a^3$$
   • Substituting the value of a, we get:
     $$V = (0.6)^3$$

4. Compute the Cube:

   • Compute (0.6 \times 0.6 \times 0.6):
     • First, (0.6 \times 0.6 = 0.36).
     • Next, (0.36 \times 0.6 = 0.216).

5. Volume in Fraction Form:

   • Convert 0.216 to a fraction:
     • (0.216 = \frac{216}{1000})
     • Simplify the fraction: ( \frac{216}{1000} = \frac{27}{125} )

6. Choose the Correct Answer:

   • The closest simplified fraction from the given options:
     • Option (A) is (\frac{8}{27})
     • Option (B) is (\frac{1}{8})
     • Option (C) is (\frac{1}{64})
     • Option (D) is (\frac{64}{27})
   • Check simplified form and decimal equivalents; neither matches immediately with ( \frac{27}{125} ).

Final Answer:

The volume of the cube is not directly provided in this form. Recheck simplification or compute finding a matching choice within the options, taking into account simplifications errors or possible miscalculations. The given approach finds the volume as a decimal (0.216) or fraction (\frac{27}{125}).