Question

1. audrey is ordering a cake for her birthday party. she needs a cake with a volume of at least 1,200 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.

1. in which of these cases will a spheres volume increase by more than 200π cm³? select two correct answers.

a. when its radius increases from 2 cm to 4 cm
b. when its radius increases from 3 cm to 5 cm
c. when its radius increases from 5 cm to 6 cm
d. when its radius increases from 6 cm to 7 cm
e. when its radius increases from 8 cm to 10 cm

Explanation:

Step1: Recall the volume formula of a sphere

The volume formula of a sphere is \(V = \frac{4}{3}\pi r^{3}\). The increase in volume \(\Delta V=\frac{4}{3}\pi(r_{2}^{3}-r_{1}^{3})\)

Step2: Calculate for option A

For \(r_1 = 2\) and \(r_2 = 4\), \(\Delta V=\frac{4}{3}\pi(4^{3}-2^{3})=\frac{4}{3}\pi(64 - 8)=\frac{4}{3}\pi\times56=\frac{224}{3}\pi\approx74.67\pi<200\pi\)

Step3: Calculate for option B

For \(r_1 = 3\) and \(r_2 = 5\), \(\Delta V=\frac{4}{3}\pi(5^{3}-3^{3})=\frac{4}{3}\pi(125 - 27)=\frac{4}{3}\pi\times98=\frac{392}{3}\pi\approx130.67\pi<200\pi\)

Step4: Calculate for option C

For \(r_1 = 5\) and \(r_2 = 6\), \(\Delta V=\frac{4}{3}\pi(6^{3}-5^{3})=\frac{4}{3}\pi(216 - 125)=\frac{4}{3}\pi\times91=\frac{364}{3}\pi\approx121.33\pi<200\pi\)

Step5: Calculate for option D

For \(r_1 = 6\) and \(r_2 = 7\), \(\Delta V=\frac{4}{3}\pi(7^{3}-6^{3})=\frac{4}{3}\pi(343 - 216)=\frac{4}{3}\pi\times127=\frac{508}{3}\pi\approx169.33\pi<200\pi\)

Step6: Calculate for option E

For \(r_1 = 8\) and \(r_2 = 10\), \(\Delta V=\frac{4}{3}\pi(10^{3}-8^{3})=\frac{4}{3}\pi(1000 - 512)=\frac{4}{3}\pi\times488=\frac{1952}{3}\pi\approx650.67\pi>200\pi\)

Answer:

E. when its radius increases from 8 cm to 10 cm