When I looked at this problem, it really scared me.
So I have no choice ... I must work on it.
-- It is given that 'y' is directly proportional to x³ .
So immediately we can write
y = k x³ .
-- For some particular value of 'x' ... call it x₀ ... y = p .
OK. So we can write
p = k x₀³
Now they want to know: What happens if x₀ gets cut in half ?
Let's see . . .
(1/2 of x₀)³ = 1/8 of x₀³ .
So let's take the last equation we wrote p = k x₀³
and divide each side by 8 :
p/8 = k 1/8 x₀³
= k (1/2 x₀)³ .
I think all this is telling us that
if y(x₀) = p
then y(x₀/2) = p/8 .
That's exactly what they're asking for.
I hope this makes sense to you. It feels reasonable to me,
but it's very hard to explain in words.
