Question

mrs. norcross bought a pack of christmas tree decorations that includes ornaments shaped like spheres and cones. which ornament takes up less space in the box?volume of a cone$v=\frac{1}{3}\pi r^{2}h$$v=\frac{1}{3}\pi r^{2}h$sphere with diameter 60 mm, cone with radius 24 mm, height 81 mm

Explanation:

Step1: Calculate sphere volume

The volume formula for a sphere is $V=\frac{4}{3}\pi r^3$. The sphere's diameter is 60 mm, so radius $r=\frac{60}{2}=30$ mm.
$$V_{sphere}=\frac{4}{3}\pi (30)^3=\frac{4}{3}\pi \times 27000=36000\pi \ \text{mm}^3$$

Step2: Calculate cone volume

The volume formula for a cone is $V=\frac{1}{3}\pi r^2 h$. The cone has radius $r=24$ mm, height $h=81$ mm.
$$V_{cone}=\frac{1}{3}\pi (24)^2 \times 81=\frac{1}{3}\pi \times 576 \times 81=15552\pi \ \text{mm}^3$$

Step3: Compare the two volumes

Compare $36000\pi$ and $15552\pi$. Since $15552\pi < 36000\pi$, the cone has a smaller volume.

Answer:

The cone-shaped ornament takes up less space, with a volume of $15552\pi$ mm³ (compared to the sphere's $36000\pi$ mm³).