Question

Solve. There's a mistake in the formula for the volume of a sphere. The radius is missing.

Answer 1

Step-by-step explanation

So we have 16 2.25in diameter balls inside a box. The volume of the empty space inside the box is equal to the volume of the box less the volume of the 16 pool balls. We have the dimensions of the balls but not those of the box. However, we can find them by looking at the picture. As you can see the balls are touching the faces of the box and the surface of contiguous balls. This means that the base of the prism is a square with a side length equal to two diameters of a pool ball. Since the diameter of each ball is 2.25 then the base of the box is a 2*2.25in=4.5in side square. Then the height of the box is 4 times the diameter of a ball so its height is 4*2.25in=9in.

Now that we have the dimensions of the box we can find its volume. This is given by the product between its base area and its height. Since its base is a square its area is given by the square of any of its sides. Then the volume of the box is:

`V_B=(4.5in)^2\cdot9in=182.25in^3`

We must also find the volume of the balls. Since the diameter of a pool ball is 2.25 in its radius is then 2.25in/2=1.125in. Then the volume of one ball is given by:

`(4)/(3)\pi r^3=(4)/(3)\pi(1.125in)^3=5.964in^3`

So the volume of 16 balls is 16 times that of one:

`V_b=16\cdot5.964in^3=95.424in^3`

Then the volume of the empty space inside the box is:

`V_B-V_b=182.25in^3-95.424in^3=86.826in^3`Answer

Since we must give the answer rounded to the nearest tenth of a cubic inch we have that the answer is 86.8in³.