y is directional to x². x is increased by 100%, find the percentage increase in y.

Given that \( y \) is directional to \( x^2 \), this means that \( y \) is proportional to \( x^2 \), which can be expressed as:

\[
y \propto x^2
\]

or

\[
y = kx^2
\]

where \( k \) is a constant.

### Step 1: Determine the initial value of \( y \)
Let the initial value of \( x \) be \( x_0 \). Then the initial value of \( y \) is:

\[
y_0 = kx_0^2
\]

### Step 2: Increase \( x \) by 100%
If \( x \) is increased by 100%, the new value of \( x \) will be:

\[
x_1 = 2x_0
\]

### Step 3: Calculate the new value of \( y \)
The new value of \( y \) when \( x = x_1 \) is:

\[
y_1 = kx_1^2 = k(2x_0)^2 = k(4x_0^2) = 4kx_0^2
\]

### Step 4: Determine the percentage increase in \( y \)
The percentage increase in \( y \) is calculated as:

\[
\text{Percentage Increase in } y = \frac{y_1 - y_0}{y_0} \times 100\%
\]

Substituting the values:

\[
\text{Percentage Increase in } y = \frac{4kx_0^2 - kx_0^2}{kx_0^2} \times 100\% = \frac{3kx_0^2}{kx_0^2} \times 100\% = 3 \times 100\% = 300\%
\]

Thus, the percentage increase in \( y \) is **300%**.

Would you like any details or have any questions?

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Here are 8 related questions to consider:
1. What happens to \( y \) if \( x \) is increased by 50%?
2. How would the percentage increase in \( y \) change if \( y \) were proportional to \( x^3 \) instead of \( x^2 \)?
3. What if \( x \) is decreased by 100%? What happens to \( y \)?
4. How can this relationship be visualized graphically?
5. What is the significance of the constant \( k \) in this relationship?
6. If \( y = kx^2 + c \), how does this affect the percentage increase in \( y \)?
7. Can you find a general formula for the percentage increase in \( y \) for any percentage increase in \( x \)?
8. What happens to \( y \) if \( x \) is increased by an arbitrary percentage, say \( p \% \)?

**Tip:** Understanding the proportional relationships in mathematical equations can help predict how changes in one variable affect another.

Understand how to calculate the percentage increase in y when x, proportional to x^2, is increased by 100%. Learn the formula and step-by-step solution to determine the impact of changes in x on y. Suitable for high school students studying proportional relationships and percentage increase in mathematics.

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