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.

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 for a sphere

The volume formula for 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 \(r_{2}=10\), \(\Delta V=\frac{4}{3}\pi(10^{3}-8^{3})=\frac{4}{3}\pi(1000 - 512)=\frac{4}{3}\pi\times488=\frac{1952}{3}\pi\approx650.67\pi>200\pi\).

Another way:
We can also use the formula \(\Delta V=\frac{4}{3}\pi(r_{2}^{3}-r_{1}^{3})\) and check \(r_{2}^{3}-r_{1}^{3}\)
For option A: \(4^{3}-2^{3}=64 - 8 = 56\)
For option B: \(5^{3}-3^{3}=125 - 27 = 98\)
For option C: \(6^{3}-5^{3}=216 - 125 = 91\)
For option D: \(7^{3}-6^{3}=343 - 216 = 127\)
For option E: \(10^{3}-8^{3}=1000 - 512 = 488\)

Since \(\Delta V=\frac{4}{3}\pi(r_{2}^{3}-r_{1}^{3})\), when \(r_{2}^{3}-r_{1}^{3}>150\) (because \(\frac{4}{3}\times150 = 200\)), \(\Delta V>200\pi\)

Answer:

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