Question

If a sphere of ice cream that was 2.5 inches in diameter was placed on top of the ice cream cone that had a height of 4 inches and a diameter of 1.75 inches and left to melt directly into the cone, would the all of the ice cream fit into the cone? Justify your answer using mathematics and a complete sentence answer.

Answer 1

Answer:

The cone would not the all of the ice cream fit into the cone.

Step-by-step explanation:

Geometrically speaking, the volume of the cone ($V_{c}$), measured in cubic inches, is described by the following equation:

$V_{c} = \frac{\pi}{3}\cdot r^{2}\cdot h$ (1)

Where:

$r$ - Base radius, measured in inches.

$h$ - Height, measured in inches.

And the ice cream ball is described by the volume equation for the sphere ($V_{s}$), measured in cubic inches:

$V_{s} = \frac{4\pi}{3} \cdot R^{3}$ (2)

Where $R$ is the radius of the ice cream ball, measured in inches.

In this case, the ice cream will fit into the cone if and only if $V_{s} \le V_{c}$. That is:

$\frac{4\pi}{3}\cdot R^{3} \le \frac{\pi}{3}\cdot r^{2}\cdot h$

$4\cdot R^{3} \le r^{2}\cdot h$

If we know that $r = 0.875\,in$, $h = 4\,in$ and $R = 1.25\,in$, then the inequation is:

$4\cdot (1.25\,in)^{3}\le (0.875\,in)^{2}\cdot (4\,in)$

$6.25\,in^{3}\le 3.063\,in^{3}$ (CRASH!)

Which leads to an absurd. Hence, the cone would not the all of the ice cream fit into the cone.