y is directly proportional to the cube of x. It is known that y = 64 for a particular value of x.
Find the value of y when x increased by 125%.

Given that \( y \) is directly proportional to the cube of \( x \), we can express this relationship as:

\[
y = kx^3
\]

where \( k \) is the constant of proportionality.

### Step 1: Find the value of \( k \)
Given \( y = 64 \) for a particular value of \( x \), let's denote this particular value of \( x \) as \( x_1 \). So:

\[
64 = kx_1^3
\]

Thus, the constant \( k \) is:

\[
k = \frac{64}{x_1^3}
\]

### Step 2: Find the new value of \( y \) when \( x \) is increased by 125%
If \( x \) is increased by 125%, the new value of \( x \) will be:

\[
x_{\text{new}} = x_1 \times \left(1 + \frac{125}{100}\right) = x_1 \times 2.25
\]

Now, substitute \( x_{\text{new}} = 2.25x_1 \) into the equation \( y = kx^3 \):

\[
y_{\text{new}} = k \times (2.25x_1)^3
\]

Using the value of \( k \) from Step 1:

\[
y_{\text{new}} = \frac{64}{x_1^3} \times (2.25x_1)^3
\]

Simplify this expression:

\[
y_{\text{new}} = 64 \times (2.25)^3
\]

### Step 3: Calculate the final value
Now, let's compute \( (2.25)^3 \):

\[
2.25^3 = 11.390625
\]

So:

\[
y_{\text{new}} = 64 \times 11.390625 = 729
\]

Therefore, the value of \( y \) when \( x \) is increased by 125% is **729**.

Would you like further details on any of these steps or have any questions?

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Here are 5 related questions for further exploration:

1. What happens to \( y \) if \( x \) is decreased by 50%?
2. How does the value of \( y \) change if \( x \) is doubled?
3. What is the new value of \( y \) if \( x \) is tripled?
4. How would the relationship change if \( y \) were proportional to the square of \( x \) instead?
5. How does the value of \( k \) influence the proportionality relationship between \( y \) and \( x^3 \)?

**Tip:** In proportionality problems, understanding the role of the constant \( k \) helps in predicting how changes in one variable affect the other.

Explore how to find the value of y when x is increased by 125% in a proportional relationship described by y = kx^3. Step-by-step solution provided with the use of proportionality and cube function formulas.

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