Ria, imagine a square of length 1 cm.

What is its area?

If the length is doubled, what happens to its area?

If the length is tripled instead, what happens to its area?

Now, imagine a cube of length 1 cm.

What is its volume?

If the length is doubled, what happens to its volume?

If the length is tripled instead, what happens to its volume?

I’m so sorry I still don’t understand

It says y is 5 so y is k 5 instead?

The actual working goes like this.

y = 5 for a particular value of x.

This does not mean that k = 5. Here, k is an unknown which we will eventually not know as well.

Since y is directly proportional to x3,
y = kx^3

For one particular value of x, y = 5, so
5 = kX^3
(I denote this particular value by X, so that in this case, x = X)

The k will remain unchanged for the rest of this question.

When this particular value of x (X) is being doubled to get 2X, our new value of Y
= kx^3
= k (2X)^3

(because x = X in the first situation and x = 2X in the new situation)

= k times 8X^3
= 8 times kX^3

But we know from earlier that kX^3 = 5 (because it was given as such, and the k is a fixed number)

= 8 times 5
= 40

The overall effect of this is that for every power of x which you see, the value of y increases by a multiplier of 2.

There are three “x” in x^3, so the multiplier 2 will take effect three times, so the overall value becomes 2^3 = 8 times that of the old value due to the three-in-a-row doubling.

Very simple.


y = kx³, k is a constant.

When x is a certain value, y = 5

So 5 = k(certain value)³ ①



Now,

when x is doubled, the right hand side becomes

k(2 × certain value)³

= k (2 × certain value) × (2 × certain value) × (2 × certain value)


= k ( 2 × 2 × 2 × (certain value) × (certain value) × (certain value) )

= k (2³ × (certain value)³ )

= k (8 × (certain value)³ )

= 8k(certain value)³

= 8 x 5 (based on equation ①)

= 40

You can try this out with some value of x.

Let's say this certain value is 7 for example.

When x = 7, y = 5

So 5 = k(7³) ①


when x is doubled, the right hand side becomes

k(2 × 7)³

= k (2 × 7) × (2 × 7) × (2 × 7)


= k ( 2 × 2 × 2 × 7 × 7 x 7)

= k (2³ × 7³)

= k (8 × 7³)

= 8k(7³)

= 8 x 5 (based on equation ①

= 40

So the idea is, if you double something, and that is cubed, the overall result is 8 times of the original. Because you are multiplying by 2 three times over.


Eg. Volume of a 3-cm cube

= 3cm x 3cm x 3cm
= 27cm³

If you double the side length to 6cm,

Volume of a 6-cm cube = 6cm x 6cm x 6cm
= 216 cm³

But this is equal to
(2 x 3cm) x (2 x 3cm) x (2 x 3cm)
= 2 x 2 x 2 x 3cm x 3cm x 3cm
= 2³ x 27cm³
= 8 x 27cm³

= 216cm³

Notice its volume is 8 times of the 3-cm cube.

Thank
You so much for all of your help it really helped your efforts and time
are greatly appreciated