Question

1. audrey is ordering a cake for her birthday party. she needs a cake with a volume of at least 1,300 cubic inches to be able to feed all her guests. determine if each cake will feed all her guests. hemisphere with a radius of 8 inches. cylinder with a diameter of 14 inches and a height of 9 inches. sphere with a diameter of 15 inches. cone with a radius of 9 inches and a height of 11 inches.

Explanation:

Step1: Volume of hemisphere

The formula for the volume of a sphere is \(V=\frac{4}{3}\pi r^{3}\). For a hemisphere, \(V_{hemisphere}=\frac{2}{3}\pi r^{3}\). Given \(r = 8\) inches, \(V_{hemisphere}=\frac{2}{3}\pi(8)^{3}=\frac{2}{3}\pi\times512=\frac{1024\pi}{3}\approx\frac{1024\times3.14}{3}\approx1071.79\) cubic inches.

Step2: Volume of cylinder

The formula for the volume of a cylinder is \(V=\pi r^{2}h\). Given \(d = 14\) inches, so \(r=\frac{d}{2}=7\) inches and \(h = 9\) inches. Then \(V=\pi(7)^{2}\times9=\pi\times49\times9 = 441\pi\approx441\times3.14 = 1384.74\) cubic inches.

Step3: Volume of sphere

Given \(d = 15\) inches, so \(r=\frac{d}{2}=7.5\) inches. Using \(V=\frac{4}{3}\pi r^{3}\), we have \(V=\frac{4}{3}\pi(7.5)^{3}=\frac{4}{3}\pi\times421.875=562.5\pi\approx562.5\times3.14 = 1766.25\) cubic inches.

Step4: Volume of cone

The formula for the volume of a cone is \(V=\frac{1}{3}\pi r^{2}h\). Given \(r = 9\) inches and \(h = 11\) inches. Then \(V=\frac{1}{3}\pi(9)^{2}\times11=\frac{1}{3}\pi\times81\times11 = 297\pi\approx297\times3.14=932.58\) cubic inches.

Answer:

• Hemisphere with a radius of 8 inches: No
• Cylinder with a diameter of 14 inches and a height of 9 inches: Yes
• Sphere with a diameter of 15 inches: Yes
• Cone with a radius of 9 inches and a height of 11 inches: No