Question

colin has a ceramic flower planter on his apartment balcony that is shaped like a cylinder. it has a height of 12 inches and a radius of 6 inches. he wants to buy another, smaller flower planter. select the planters that have less volume than colins ceramic planter. a planter shaped like a cone with a height of 12 inches and a diameter of 12 inches. a planter shaped like a sphere with a diameter of 12 inches. a planter shaped like a cylinder with a height of 12 inches and a diameter of 10 inches. a planter shaped like a cylinder with a height of 6 inches and a diameter of 12 inches.

Explanation:

Step1: Calculate the volume of Colin's cylindrical planter

The formula for the volume of a cylinder is \(V = \pi r^{2}h\). Given \(r = 6\) inches and \(h=12\) inches.
\(V_{cylinder}=\pi\times6^{2}\times12=\pi\times36\times 12 = 432\pi\) cubic inches.

Step2: Calculate the volume of the conical planter

The formula for the volume of a cone is \(V=\frac{1}{3}\pi r^{2}h\). Given diameter \(d = 12\) inches, so \(r=\frac{d}{2}=6\) inches and \(h = 12\) inches.
\(V_{cone}=\frac{1}{3}\pi\times6^{2}\times12=\frac{1}{3}\pi\times36\times12 = 144\pi\) cubic inches. Since \(144\pi<432\pi\).

Step3: Calculate the volume of the spherical planter

The formula for the volume of a sphere is \(V=\frac{4}{3}\pi r^{3}\). Given diameter \(d = 12\) inches, so \(r = 6\) inches.
\(V_{sphere}=\frac{4}{3}\pi\times6^{3}=\frac{4}{3}\pi\times216 = 288\pi\) cubic inches. Since \(288\pi<432\pi\).

Step4: Calculate the volume of the first alternative cylindrical planter

The formula for the volume of a cylinder is \(V=\pi r^{2}h\). Given diameter \(d = 10\) inches, so \(r = 5\) inches and \(h = 12\) inches.
\(V_{cylinder1}=\pi\times5^{2}\times12=\pi\times25\times12=300\pi\) cubic inches. Since \(300\pi<432\pi\).

Step5: Calculate the volume of the second alternative cylindrical planter

The formula for the volume of a cylinder is \(V=\pi r^{2}h\). Given diameter \(d = 12\) inches, so \(r = 6\) inches and \(h = 6\) inches.
\(V_{cylinder2}=\pi\times6^{2}\times6=\pi\times36\times6=216\pi\) cubic inches. Since \(216\pi<432\pi\).

Answer:

A. A planter shaped like a cone with a height of 12 inches and a diameter of 12 inches.
B. A planter shaped like a sphere with a diameter of 12 inches.
C. A planter shaped like a cylinder with a height of 12 inches and a diameter of 10 inches.
D. A planter shaped like a cylinder with a height of 6 inches and a diameter of 12 inches.