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Answer :
To solve this problem, we need to understand what it means for [tex]\( y \)[/tex] to vary inversely as [tex]\( x \)[/tex].
When [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex], it means that the product of [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is a constant. We can express this relationship as:
[tex]\[ y \times x = k \][/tex]
Here, [tex]\( k \)[/tex] is the constant of variation.
1. Finding the Constant [tex]\( k \)[/tex]:
- We are given that [tex]\( y = 50 \)[/tex] when [tex]\( x = 10 \)[/tex].
- Substitute these values into the inverse variation formula to find [tex]\( k \)[/tex]:
[tex]\[ 50 \times 10 = k \][/tex]
[tex]\[ k = 500 \][/tex]
2. Finding the New Value of [tex]\( y \)[/tex] When [tex]\( x = 20 \)[/tex]:
- We need to determine the value of [tex]\( y \)[/tex] when [tex]\( x = 20 \)[/tex].
- Since [tex]\( y \times x = k \)[/tex], substitute [tex]\( x = 20 \)[/tex] and [tex]\( k = 500 \)[/tex] into the equation to solve for [tex]\( y \)[/tex]:
[tex]\[ y \times 20 = 500 \][/tex]
[tex]\[ y = \frac{500}{20} \][/tex]
[tex]\[ y = 25 \][/tex]
Therefore, the value of [tex]\( y \)[/tex] when [tex]\( x = 20 \)[/tex] is 25.
When [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex], it means that the product of [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is a constant. We can express this relationship as:
[tex]\[ y \times x = k \][/tex]
Here, [tex]\( k \)[/tex] is the constant of variation.
1. Finding the Constant [tex]\( k \)[/tex]:
- We are given that [tex]\( y = 50 \)[/tex] when [tex]\( x = 10 \)[/tex].
- Substitute these values into the inverse variation formula to find [tex]\( k \)[/tex]:
[tex]\[ 50 \times 10 = k \][/tex]
[tex]\[ k = 500 \][/tex]
2. Finding the New Value of [tex]\( y \)[/tex] When [tex]\( x = 20 \)[/tex]:
- We need to determine the value of [tex]\( y \)[/tex] when [tex]\( x = 20 \)[/tex].
- Since [tex]\( y \times x = k \)[/tex], substitute [tex]\( x = 20 \)[/tex] and [tex]\( k = 500 \)[/tex] into the equation to solve for [tex]\( y \)[/tex]:
[tex]\[ y \times 20 = 500 \][/tex]
[tex]\[ y = \frac{500}{20} \][/tex]
[tex]\[ y = 25 \][/tex]
Therefore, the value of [tex]\( y \)[/tex] when [tex]\( x = 20 \)[/tex] is 25.
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