To find the value of [tex]\( y \)[/tex] when [tex]\( x = 30 \)[/tex] given that [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], let's break down the problem step-by-step.
1. Understand Direct Variation:
- When [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], the relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] can be written as:
[tex]\[
y = kx
\][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.
2. Given Information:
- We are given that [tex]\( y = 2 \)[/tex] when [tex]\( x = 10 \)[/tex].
3. Find the Constant of Proportionality [tex]\( k \)[/tex]:
- Substitute the given values into the direct variation equation to solve for [tex]\( k \)[/tex]:
[tex]\[
2 = k \cdot 10
\][/tex]
[tex]\[
k = \frac{2}{10} = 0.2
\][/tex]
4. Use the Constant to Find [tex]\( y \)[/tex] for [tex]\( x = 30 \)[/tex]:
- Now that we have the constant of proportionality [tex]\( k = 0.2 \)[/tex], we can use it to find [tex]\( y \)[/tex] when [tex]\( x = 30 \)[/tex]. Substitute [tex]\( k \)[/tex] and [tex]\( x = 30 \)[/tex] into the direct variation equation:
[tex]\[
y = 0.2 \cdot 30
\][/tex]
[tex]\[
y = 6.0
\][/tex]
Therefore, when [tex]\( x \)[/tex] is 30, the value of [tex]\( y \)[/tex] is [tex]\( 6.0 \)[/tex].
