Given that 𝑦 squared is
proportional to 𝑥 cubed, where 𝑦 equals negative 15 when 𝑥 equals seven, find the
relation between 𝑥 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 𝑥 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 𝑘.
So we can say that 𝑦 squared is
equal to 𝑘 multiplied by 𝑥 cubed, where 𝑘 is just a constant, and it’s known as
the proportionality constant. So therefore, looking at the
question to find the relation between 𝑥 and 𝑦, what we need to do is actually find
the value of 𝑘, because once we found the value of 𝑘, we can rewrite this to give
us a relationship between 𝑥 and 𝑦.
So now what we need to do to
actually find 𝑘 is substitute in the values of 𝑦 and 𝑥 that we have. So we have 𝑦 is equal to negative
15 and 𝑥 is equal to seven. So therefore, when we actually
substitute these values in, what we get is that negative 15 squared is equal to 𝑘
multiplied by seven cubed. So then when we actually simplify,
what we’re gonna get is 225 is equal to 343𝑘.
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
Okay, great! So as I said, we’ve got 225 equals
343𝑘. So now what we do is actually
divide each side of the equation by 343, cause like we said we want to find 𝑘. So we get 225 over 343 is equal to
𝑘. So now what we’re gonna do is
actually substitute our value of 𝑘 into 𝑦 squared equals 𝑘 multiplied by 𝑥 cubed
to give us our relation between 𝑥 and 𝑦. So therefore, when we do that, we
can say that, given that 𝑦 squared is proportional to 𝑥 cubed, where 𝑦 equals
negative 15 when 𝑥 equals seven, the relation between 𝑥 and 𝑦 is 𝑦 squared is
equal to 225 over 343𝑥 cubed.