Question Video: Finding Values Using a Direct Variation Relationship

Video Transcript

Given that 𝑥 is proportional to 𝑦 cubed and 𝑥 equals 81 when 𝑦 is equal to three, what is 𝑥 when 𝑦 is equal to four?

In this question, we’re dealing with direct proportion or variation. We are told that 𝑥 varies directly with 𝑦 cubed. This can be rewritten as the equation 𝑥 is equal to the constant 𝑘 multiplied by 𝑦 cubed. Dividing both sides of this equation by 𝑦 cubed gives us the constant 𝑘 is equal to 𝑥 divided by 𝑦 cubed. We are also told that when 𝑥 is equal to 81, 𝑦 is equal to three. This means that we can calculate the value of 𝑘 by dividing 81 by three cubed. Three cubed is equal to 27, as three multiplied by three is nine and multiplying this by three gives us 27. This means that 𝑘 is equal to 81 divided by 27. There are three 27s in 81. Therefore, 𝑘 is equal to three.

If we didn’t spot this, we could have firstly canceled the fraction by dividing the numerator and denominator by nine. This would leave us with nine divided by three, which we know is equal to three. Alternatively, we might have noticed that 81 is equal to three to the fourth power. And dividing this by three to the third power or three cubed would give us three to the power of one, which is three. Substituting this value of 𝑘 back into our equation gives us 𝑥 is equal to three 𝑦 cubed. We now need to calculate the value of 𝑥 when 𝑦 is equal to four. This gives us 𝑥 is equal to three multiplied by four cubed. Four cubed is equal to 64. And multiplying this by three gives us 192. If 𝑥 is proportional to 𝑦 cubed and 𝑥 equals 81 when 𝑦 is equal to three, then 𝑥 is equal to 192 when 𝑦 is equal to four.