Answer :
Certainly! Let's solve each of these inverse variation problems step by step:
### 4. Find [tex]\( y \)[/tex] when [tex]\( x = -9 \)[/tex] if [tex]\( y = 3 \)[/tex] when [tex]\( x = 6 \)[/tex].
In an inverse variation, the product [tex]\( xy \)[/tex] is constant.
1. Firstly, find the constant of variation [tex]\( k \)[/tex] using the given values:
[tex]\[ k = x_1 y_1 = 6 \cdot 3 = 18 \][/tex]
2. Now use this constant to find the new value of [tex]\( y_2 \)[/tex] when [tex]\( x_2 = -9 \)[/tex]:
[tex]\[ k = x_2 y_2 \implies 18 = -9 y_2 \implies y_2 = \frac{18}{-9} = -2 \][/tex]
So, [tex]\( y = -2 \)[/tex] when [tex]\( x = -9 \)[/tex].
### 5. Suppose [tex]\( y = 5 \)[/tex] when [tex]\( x = 16 \)[/tex]. Find [tex]\( x \)[/tex] when [tex]\( y = 10 \)[/tex].
1. Find the constant [tex]\( k \)[/tex]:
[tex]\[ k = x_1 y_1 = 16 \cdot 5 = 80 \][/tex]
2. Now use the constant to find the new value of [tex]\( x_2 \)[/tex] when [tex]\( y_2 = 10 \)[/tex]:
[tex]\[ k = x_2 y_2 \implies 80 = x_2 \cdot 10 \implies x_2 = \frac{80}{10} = 8 \][/tex]
So, [tex]\( x = 8 \)[/tex] when [tex]\( y = 10 \)[/tex].
### 6. If [tex]\( y = 4.5 \)[/tex] when [tex]\( x = 6 \)[/tex], find [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex].
1. Find the constant [tex]\( k \)[/tex]:
[tex]\[ k = x_1 y_1 = 6 \cdot 4.5 = 27 \][/tex]
2. Use the constant to find the new value of [tex]\( y_2 \)[/tex] when [tex]\( x_2 = 3 \)[/tex]:
[tex]\[ k = x_2 y_2 \implies 27 = 3 y_2 \implies y_2 = \frac{27}{3} = 9 \][/tex]
So, [tex]\( y = 9 \)[/tex] when [tex]\( x = 3 \)[/tex].
### 7. Find [tex]\( y \)[/tex] when [tex]\( x = 0.125 \)[/tex] if [tex]\( y = 1.5 \)[/tex] when [tex]\( x = 2.5 \)[/tex].
1. Find the constant [tex]\( k \)[/tex]:
[tex]\[ k = x_1 y_1 = 2.5 \cdot 1.5 = 3.75 \][/tex]
2. Use the constant to find the new value of [tex]\( y_2 \)[/tex] when [tex]\( x_2 = 0.125 \)[/tex]:
[tex]\[ k = x_2 y_2 \implies 3.75 = 0.125 y_2 \implies y_2 = \frac{3.75}{0.125} = 30 \][/tex]
So, [tex]\( y = 30 \)[/tex] when [tex]\( x = 0.125 \)[/tex].
### 8. Find [tex]\( x \)[/tex] when [tex]\( y = 0.9 \)[/tex], if [tex]\( y = 1.5 \)[/tex] when [tex]\( x = 0.3 \)[/tex].
1. Find the constant [tex]\( k \)[/tex]:
[tex]\[ k = x_1 y_1 = 0.3 \cdot 1.5 = 0.45 \][/tex]
2. Use the constant to find the new value of [tex]\( x_2 \)[/tex] when [tex]\( y_2 = 0.9 \)[/tex]:
[tex]\[ k = x_2 y_2 \implies 0.45 = x_2 \cdot 0.9 \implies x_2 = \frac{0.45}{0.9} = 0.5 \][/tex]
So, [tex]\( x = 0.5 \)[/tex] when [tex]\( y = 0.9 \)[/tex].
### 9. Suppose [tex]\( x = -8 \)[/tex] when [tex]\( y = 6 \)[/tex]. Find [tex]\( y \)[/tex] when [tex]\( x = 16 \)[/tex].
1. Find the constant [tex]\( k \)[/tex]:
[tex]\[ k = x_1 y_1 \implies -8 \cdot 6 = -48 \][/tex]
2. Use the constant to find the new value of [tex]\( y_2 \)[/tex] when [tex]\( x_2 = 16 \)[/tex]:
[tex]\[ k = x_2 y_2 \implies -48 = 16 y_2 \implies y_2 = \frac{-48}{16} = -3 \][/tex]
So, [tex]\( y = -3 \)[/tex] when [tex]\( x = 16 \)[/tex].
### 10. If [tex]\( y = \frac{1}{4} \)[/tex] when [tex]\( x = 8 \)[/tex], find [tex]\( x \)[/tex] when [tex]\( y = \frac{2}{5} \)[/tex].
1. Find the constant [tex]\( k \)[/tex]:
[tex]\[ k = x_1 y_1 \implies 8 \cdot \frac{1}{4} = 2 \][/tex]
2. Use the constant to find the new value of [tex]\( x_2 \)[/tex] when [tex]\( y_2 = \frac{2}{5} \)[/tex]:
[tex]\[ k = x_2 y_2 \implies 2 = x_2 \cdot \frac{2}{5} \implies x_2 = \frac{2}{\frac{2}{5}} = 2 \cdot \frac{5}{2} = 5 \][/tex]
So, [tex]\( x = 5 \)[/tex] when [tex]\( y = \frac{2}{5} \)[/tex].
### 4. Find [tex]\( y \)[/tex] when [tex]\( x = -9 \)[/tex] if [tex]\( y = 3 \)[/tex] when [tex]\( x = 6 \)[/tex].
In an inverse variation, the product [tex]\( xy \)[/tex] is constant.
1. Firstly, find the constant of variation [tex]\( k \)[/tex] using the given values:
[tex]\[ k = x_1 y_1 = 6 \cdot 3 = 18 \][/tex]
2. Now use this constant to find the new value of [tex]\( y_2 \)[/tex] when [tex]\( x_2 = -9 \)[/tex]:
[tex]\[ k = x_2 y_2 \implies 18 = -9 y_2 \implies y_2 = \frac{18}{-9} = -2 \][/tex]
So, [tex]\( y = -2 \)[/tex] when [tex]\( x = -9 \)[/tex].
### 5. Suppose [tex]\( y = 5 \)[/tex] when [tex]\( x = 16 \)[/tex]. Find [tex]\( x \)[/tex] when [tex]\( y = 10 \)[/tex].
1. Find the constant [tex]\( k \)[/tex]:
[tex]\[ k = x_1 y_1 = 16 \cdot 5 = 80 \][/tex]
2. Now use the constant to find the new value of [tex]\( x_2 \)[/tex] when [tex]\( y_2 = 10 \)[/tex]:
[tex]\[ k = x_2 y_2 \implies 80 = x_2 \cdot 10 \implies x_2 = \frac{80}{10} = 8 \][/tex]
So, [tex]\( x = 8 \)[/tex] when [tex]\( y = 10 \)[/tex].
### 6. If [tex]\( y = 4.5 \)[/tex] when [tex]\( x = 6 \)[/tex], find [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex].
1. Find the constant [tex]\( k \)[/tex]:
[tex]\[ k = x_1 y_1 = 6 \cdot 4.5 = 27 \][/tex]
2. Use the constant to find the new value of [tex]\( y_2 \)[/tex] when [tex]\( x_2 = 3 \)[/tex]:
[tex]\[ k = x_2 y_2 \implies 27 = 3 y_2 \implies y_2 = \frac{27}{3} = 9 \][/tex]
So, [tex]\( y = 9 \)[/tex] when [tex]\( x = 3 \)[/tex].
### 7. Find [tex]\( y \)[/tex] when [tex]\( x = 0.125 \)[/tex] if [tex]\( y = 1.5 \)[/tex] when [tex]\( x = 2.5 \)[/tex].
1. Find the constant [tex]\( k \)[/tex]:
[tex]\[ k = x_1 y_1 = 2.5 \cdot 1.5 = 3.75 \][/tex]
2. Use the constant to find the new value of [tex]\( y_2 \)[/tex] when [tex]\( x_2 = 0.125 \)[/tex]:
[tex]\[ k = x_2 y_2 \implies 3.75 = 0.125 y_2 \implies y_2 = \frac{3.75}{0.125} = 30 \][/tex]
So, [tex]\( y = 30 \)[/tex] when [tex]\( x = 0.125 \)[/tex].
### 8. Find [tex]\( x \)[/tex] when [tex]\( y = 0.9 \)[/tex], if [tex]\( y = 1.5 \)[/tex] when [tex]\( x = 0.3 \)[/tex].
1. Find the constant [tex]\( k \)[/tex]:
[tex]\[ k = x_1 y_1 = 0.3 \cdot 1.5 = 0.45 \][/tex]
2. Use the constant to find the new value of [tex]\( x_2 \)[/tex] when [tex]\( y_2 = 0.9 \)[/tex]:
[tex]\[ k = x_2 y_2 \implies 0.45 = x_2 \cdot 0.9 \implies x_2 = \frac{0.45}{0.9} = 0.5 \][/tex]
So, [tex]\( x = 0.5 \)[/tex] when [tex]\( y = 0.9 \)[/tex].
### 9. Suppose [tex]\( x = -8 \)[/tex] when [tex]\( y = 6 \)[/tex]. Find [tex]\( y \)[/tex] when [tex]\( x = 16 \)[/tex].
1. Find the constant [tex]\( k \)[/tex]:
[tex]\[ k = x_1 y_1 \implies -8 \cdot 6 = -48 \][/tex]
2. Use the constant to find the new value of [tex]\( y_2 \)[/tex] when [tex]\( x_2 = 16 \)[/tex]:
[tex]\[ k = x_2 y_2 \implies -48 = 16 y_2 \implies y_2 = \frac{-48}{16} = -3 \][/tex]
So, [tex]\( y = -3 \)[/tex] when [tex]\( x = 16 \)[/tex].
### 10. If [tex]\( y = \frac{1}{4} \)[/tex] when [tex]\( x = 8 \)[/tex], find [tex]\( x \)[/tex] when [tex]\( y = \frac{2}{5} \)[/tex].
1. Find the constant [tex]\( k \)[/tex]:
[tex]\[ k = x_1 y_1 \implies 8 \cdot \frac{1}{4} = 2 \][/tex]
2. Use the constant to find the new value of [tex]\( x_2 \)[/tex] when [tex]\( y_2 = \frac{2}{5} \)[/tex]:
[tex]\[ k = x_2 y_2 \implies 2 = x_2 \cdot \frac{2}{5} \implies x_2 = \frac{2}{\frac{2}{5}} = 2 \cdot \frac{5}{2} = 5 \][/tex]
So, [tex]\( x = 5 \)[/tex] when [tex]\( y = \frac{2}{5} \)[/tex].
