𝑦 varies inversely with 𝑥 cubed. The constant of proportionality is equal to six. Which of the following equations represents this relation? Is it a) 𝑦 equals six divided by 𝑥 cubed, b) 𝑦 equals six divided by 𝑥, c) 𝑦 equals six minus 𝑥 cubed, or d) 𝑦 equals six 𝑥 cubed?
As 𝑦 varies inversely with 𝑥 cubed, we can say that 𝑦 is proportional to one divided by 𝑥 cubed or 𝑦 is proportional to one over 𝑥 cubed. Replacing our proportion symbol with a constant of proportionality gives us the equation 𝑦 equals 𝑘 divided by 𝑥 cubed.
As the constant of proportionality in this question is equal to six, we can write 𝑦 equals six divided by 𝑥 cubed. Therefore, the correct answer in this case is a: 𝑦 equals six divided by 𝑥 cubed.
As an aside, answer b 𝑦 equals six divided by 𝑥 would be the answer if 𝑦 was varying inversely with 𝑥. In the same way answer d 𝑦 equals six 𝑥 cubed would be the answer if 𝑦 was varying directly with 𝑥 cubed. Answer c does not demonstrate a direct or an inverse proportion between 𝑥 and 𝑦.