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If [tex]y[/tex] varies directly as [tex]x[/tex], and [tex]y[/tex] is 6 when [tex]x[/tex] is 72, what is the value of [tex]y[/tex] when [tex]x[/tex] is 8?

A. [tex]\frac{1}{9}[/tex]
B. [tex]\frac{2}{3}[/tex]
C. 54
D. 96


Sagot :

To solve the problem given that [tex]\( y \)[/tex] varies directly as [tex]\( x \)[/tex], we need to follow the concept of direct variation, which means [tex]\( y = kx \)[/tex] for some constant [tex]\( k \)[/tex].

Here’s the step-by-step approach to solving the problem:

1. Determine the constant of variation [tex]\( k \)[/tex]:
- We are given that [tex]\( y = 6 \)[/tex] when [tex]\( x = 72 \)[/tex].
- Since [tex]\( y \)[/tex] varies directly as [tex]\( x \)[/tex], we can express this relationship as:
[tex]\[ y = kx \][/tex]
- Substitute the given values into the equation to solve for [tex]\( k \)[/tex]:
[tex]\[ 6 = k \cdot 72 \][/tex]
- Solving for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{6}{72} = \frac{1}{12} \][/tex]

2. Use the constant [tex]\( k \)[/tex] to find [tex]\( y \)[/tex] when [tex]\( x = 8 \)[/tex]:
- Now that we have [tex]\( k = \frac{1}{12} \)[/tex], we use this to find the new value of [tex]\( y \)[/tex] for [tex]\( x = 8 \)[/tex].
- Substitute [tex]\( k \)[/tex] and [tex]\( x = 8 \)[/tex] back into the direct variation equation [tex]\( y = kx \)[/tex]:
[tex]\[ y = \frac{1}{12} \cdot 8 \][/tex]
- Simplify the expression:
[tex]\[ y = \frac{8}{12} = \frac{2}{3} \][/tex]

So the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is 8 is [tex]\( \frac{2}{3} \)[/tex].

Therefore, the correct answer is:
[tex]\[ \boxed{\frac{2}{3}} \][/tex]
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