If the volume of the figure formed is 256 cu. cm, find the height of each pyramid

Question

Six square based pyramids fit exactly not the 6 faces of cube with sides of 4cm. please show work and explain 

Answer 1

I'm guessing you meant "...fit exactly ON the 6 faces..."

 

I'm also guessing that the heights of all the pyramids are the same.  Is that true?  Let's assume it is, because if it isn't, there's no way to solve the problem without more information about the figure.

 

Because the sides of the cube are 4 cm, the volume of the cube is 4³ cm³, or 64 cm³.

 

The formula for the volume of a pyramid is 1/3 A h, where A is the area of the base and h is the height.

 

The way you describe the figure, the area of the base is just the area of a face of the cube.  That's 4*4, or 16 cm².

 

The height is what we're trying to figure out, so let's just call it h.

 

So the volume of each pyramid is 1/3 * 16 * h = 16/3 h.

 

Since there are 6 of them, total volume of the pyramids is 6 * 16/3 h = 32h.

 

We already know that the volume of the cube is 64.

 

So the total volume of the figure is 32h + 64, and that's equal to 256.

 

So solve the equation: 32h + 64 = 256

 

Subtract 64 from both sides:

32h = 192

 

Divide both sides by 32:

h = 6.

 

So the answer is that the height is 6 cm.