Question

Aaron and Nadia are making miniature pinatas. They're evenly splitting a bag of candy between them, so they want their pinatas to hold the same amount of candy. Aaron's pinata is a sphere with a radius of 3 inches. If Nadia wants to make her pinata a cone with the same radius, how tall should it be?

Answer 1

Answer:

4 inches

Step-by-step explanation:

To find the height of Nadia's cone pinata, we can use the formula for the volume of a cone:

$\Large\boxed{\boxed{V_{\textsf{cone}} = \dfrac{1}{3} \pi r_{\textsf{cone}}^2 h_{\textsf{cone}} }}$

where:

• $V_{\textsf{cone}}$ is the volume of the cone (which needs to be the same as the volume of Aaron's sphere),
• $\pi$ is a constant (approximately 3.14159),
• $r_{\textsf{cone}}$ is the radius of Nadia's cone,
• $h_{\textsf{cone}}$ is the height of Nadia's cone.

The volume of Aaron's sphere is given by the formula:

$\Large\boxed{\boxed{V_{\textsf{sphere}} = \dfrac{4}{3} \pi r_{\textsf{sphere}}^3 }}$

where:

• $V_{\textsf{sphere}}$ is the volume of the sphere,
• $r_{\textsf{sphere}}$ is the radius of the sphere.

Since Aaron and Nadia are splitting the candy evenly, the volumes of their pinatas should be equal:

$V_{\textsf{cone}} = V_{\textsf{sphere}}$

Substituting in the formulas, we get:

$\dfrac{1}{3} \pi r_{\textsf{cone}}^2 h_{\textsf{cone}} = \dfrac{4}{3} \pi r_{\textsf{sphere}}^3$

Aaron's sphere has a radius $r_{\textsf{sphere}}$ of 3 inches.

So, substituting $r_{\textsf{sphere}} = 3$, we get:

$\dfrac{1}{3} \pi r_{\textsf{cone}}^2 h_{\textsf{cone}} = \dfrac{4}{3} \pi (3)^3$

Now, we can solve for $h_{\textsf{cone}}$:

$\dfrac{1}{\cancel{3}} \cancel{\pi} r_{\textsf{cone}}^2 h_{\textsf{cone}} = \dfrac{4}{\cancel{3}}\cancel{ \pi} (3)^3$

$r_{\textsf{cone}}^2 h_{\textsf{cone}} = 4(3)^2$

$r_{\textsf{cone}}^2 h_{\textsf{cone}} = 36$

$h_{\textsf{cone}} = \dfrac{36}{r_{\textsf{cone}}^2}$

Since $r_{\textsf{cone}}$ is the same as the radius of Aaron's sphere (3 inches):

$h_{\textsf{cone}} = \dfrac{36}{3^2}$

$h_{\textsf{cone}} = \dfrac{36}{9}$

$h_{\textsf{cone}} = 4$

Therefore, Nadia should make her cone pinata with a height ($h_{\textsf{cone}}$) of 4 inches.