Video: Finding the Proportional Relationship between Two Quantities with Direct Variation

Given that 𝑦² ∝ π‘₯Β³, where 𝑦 = βˆ’15 when π‘₯ = 7, find the relation between π‘₯ and 𝑦.

03:14

Video Transcript

Given that 𝑦 squared is proportional to π‘₯ cubed, where 𝑦 equals negative 15 when π‘₯ equals seven, find the relation between π‘₯ and 𝑦.

So I have a look at the question. We can see that actually, with this expression, we have that 𝑦 squared is proportional to π‘₯ cubed. However, in this form, it’s not much of use to us because we actually we can’t do much with it. So therefore, what we can do is rewrite this as an equation by introducing π‘˜.

So we can say that 𝑦 squared is equal to π‘˜ multiplied by π‘₯ cubed, where π‘˜ is just a constant, and it’s known as the proportionality constant. So therefore, looking at the question to find the relation between π‘₯ and 𝑦, what we need to do is actually find the value of π‘˜, because once we found the value of π‘˜, we can rewrite this to give us a relationship between π‘₯ and 𝑦.

So now what we need to do to actually find π‘˜ is substitute in the values of 𝑦 and π‘₯ that we have. So we have 𝑦 is equal to negative 15 and π‘₯ is equal to seven. So therefore, when we actually substitute these values in, what we get is that negative 15 squared is equal to π‘˜ multiplied by seven cubed. So then when we actually simplify, what we’re gonna get is 225 is equal to 343π‘˜.

Now let’s look at how we got those values. Well, we had negative 15 all squared, but a negative multiplied by a negative is a positive. And we know that 15 multiplied by 15 gives us 225, so we get positive 225.

And now to actually work out 343, well, this might be one that you don’t actually know, so let’s see if we could actually calculate it. Well, we know that seven cubed is equal to seven multiplied by seven multiplied by seven. Well, this is actually gonna be equal to 49 multiplied by seven, and that’s because we have seven multiplied by seven is 49, then multiplied by seven.

And what we’re gonna do is use the column method to actually multiply this. So I’m gonna start with seven multiplied by nine. Well, seven nines are 63, so what we do is we put three in the units column and carry the six. And next, we have seven multiplied by four. Well, seven multiplied by four is 28. Well, 28 add the six that we carried is gonna give us 34. So therefore, we’re gonna have a four in the tens column and a three in the hundreds column. So we can say that 49 multiplied by seven is 343. So therefore, seven cubed is 343.

Okay, great! So as I said, we’ve got 225 equals 343π‘˜. So now what we do is actually divide each side of the equation by 343, cause like we said we want to find π‘˜. So we get 225 over 343 is equal to π‘˜. So now what we’re gonna do is actually substitute our value of π‘˜ into 𝑦 squared equals π‘˜ multiplied by π‘₯ cubed to give us our relation between π‘₯ and 𝑦. So therefore, when we do that, we can say that, given that 𝑦 squared is proportional to π‘₯ cubed, where 𝑦 equals negative 15 when π‘₯ equals seven, the relation between π‘₯ and 𝑦 is 𝑦 squared is equal to 225 over 343π‘₯ cubed.

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