To solve this problem, we'll go step by step.
1. Understand the relationship:
- The problem states that [tex]\( y \)[/tex] is directly proportional to [tex]\( x^2 \)[/tex]. Mathematically, this can be written as:
[tex]\[
y = k \cdot x^2
\][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.
2. Find the constant of proportionality [tex]\( k \)[/tex]:
- We're given that [tex]\( y = 10 \)[/tex] when [tex]\( x = 2 \)[/tex]. Substituting these values into the equation, we get:
[tex]\[
10 = k \cdot 2^2
\][/tex]
- Simplify the equation to solve for [tex]\( k \)[/tex]:
[tex]\[
10 = k \cdot 4
\][/tex]
[tex]\[
k = \frac{10}{4}
\][/tex]
[tex]\[
k = 2.5
\][/tex]
3. Double the value of [tex]\( x \)[/tex]:
- We need to find the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is doubled. If [tex]\( x \)[/tex] is initially 2, then doubling it gives:
[tex]\[
x_{\text{new}} = 2 \times 2 = 4
\][/tex]
4. Calculate the new value of [tex]\( y \)[/tex]:
- Substitute [tex]\( x_{\text{new}} = 4 \)[/tex] back into the proportionality equation:
[tex]\[
y_{\text{new}} = k \cdot x_{\text{new}}^2
\][/tex]
- Using [tex]\( k = 2.5 \)[/tex] and [tex]\( x_{\text{new}} = 4 \)[/tex]:
[tex]\[
y_{\text{new}} = 2.5 \cdot 4^2
\][/tex]
[tex]\[
y_{\text{new}} = 2.5 \cdot 16
\][/tex]
[tex]\[
y_{\text{new}} = 40
\][/tex]
Hence, when the value of [tex]\( x \)[/tex] is doubled, the value of [tex]\( y \)[/tex] becomes 40.
