High School

One cubic meter of concrete has a mass of [tex]2.30 \times 10^3 \text{ kg}[/tex], and 1.00 cubic meter of iron has a mass of [tex]7.86 \times 10^3 \text{ kg}[/tex].

Find the radius of a concrete sphere whose mass is the same as that of an iron sphere with a radius of [tex]1.57 \text{ cm}[/tex].

Answer :

Final answer:

To find the radius of the concrete sphere with the same mass as an iron sphere, set up an equation equating the volumes of the two spheres. Solve for the radius of the concrete sphere using the equation (4/3)πr_c^3 = 1.00 m^3.

Explanation:

To find the radius of a concrete sphere with the same mass as an iron sphere, we need to compare the masses of the two spheres and use the formula for the volume of a sphere. The mass of the concrete sphere is given as 2.30 x 10³ kg per cubic meter, while the mass of the iron sphere is given as 7.86 x 10³ kg per cubic meter. We can set up an equation equating the volumes of the two spheres and solve for the radius of the concrete sphere.

Let's assume the radius of the concrete sphere is rc.

For the iron sphere:

  • Volume = (4/3)πr3 = 1.00 m3
  • Mass = density x volume = 7.86 x 10³ kg

For the concrete sphere:

  • Volume = (4/3)πrc3 = 1.00 m3
  • Mass = density x volume = 2.30 x 10³ kg

Setting the masses equal:

  • 7.86 x 10³ kg = 2.30 x 10³ kg

Now we can solve for the radius of the concrete sphere:

  • (4/3)πrc3 = 1.00 m3

Rearranging the equation:

  • rc3 = 0.75 / π
  • rc = (0.75 / π)1/3

Using this equation, we can calculate the radius of the concrete sphere.

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