Question

aviannas reasoning we should buy the cone - shaped candy because it has a larger volume. since the radii of the cone and sphere are the same, but the cone has a height of 12, its volume is bigger. we can buy either candy because their volumes are the same. the radii of the cone and the sphere are the same, but the cone has a height that is exactly 4 times its radius so the volumes are equal. which statement about aviannas and aviannas reasoning is true? - avianna is correct. the volume of the cone - shaped candy is larger. - avianna is correct. the volumes of the two candies are the same. - neither student is correct. the volume of the sphere - shaped candy is larger. - both students are correct.

Explanation:

Step1: Define given values

Let the radius of both cone and sphere be $r$. For the cone, height $h = 12$, and $h=4r$, so $r=\frac{12}{4}=3$.

Step2: Calculate cone volume

Use cone volume formula $V_{cone}=\frac{1}{3}\pi r^2 h$.
$V_{cone}=\frac{1}{3}\pi (3)^2 (12)=\frac{1}{3}\pi \times9\times12=36\pi$

Step3: Calculate sphere volume

Use sphere volume formula $V_{sphere}=\frac{4}{3}\pi r^3$.
$V_{sphere}=\frac{4}{3}\pi (3)^3=\frac{4}{3}\pi \times27=36\pi$

Step4: Compare volumes

$V_{cone}=V_{sphere}=36\pi$, so volumes are equal.

Answer:

Aviana is correct. The volumes of the two candies are the same.