# Suppose that x and y vary inversely and that y = 8/3 when x = 8. Write a function that models the...

## Question:

Suppose that x and y vary inversely and that y = 8/3 when x = 8. Write a function that models the inverse variation and find y when x = 4.

## Inverse Variation:

• Variation: The two types of variation between variables are direct variation and inverse variation.
• Direct Variation: Direct variation explains a relationship between two variables that are directly proportional or change in the same direction.
• Inverse variation: Inverse variation describes a relationship between one dependent variable and two or more other independent variables that change in the opposite direction as the dependent variable.

If variables {eq}\rm x\; and\; y {/eq} vary inversely, we can write the relationship as:

• {eq}x\propto \dfrac{1}{y} {/eq}

Removing the proportionality sign and adding a proportionaliy constant, we get:

• {eq}x = \dfrac{k}{y} {/eq}

If {eq}y = \dfrac{8}{3} {/eq} when {eq}x = 8 {/eq}, then:

• {eq}8 = \dfrac{k}{\dfrac{8}{3}} {/eq}

Solving for k:

• {eq}k = \dfrac{8}{3}\times 8 = \dfrac{64}{3} {/eq}

Therefore, the equation showing the relationship between the two variables is:

• {eq}x = \dfrac{64}{3y} {/eq}

Using the above equation, the value of {eq}y {/eq} when {eq}x = 4 {/eq} is:

• {eq}4 = \dfrac{64}{3y} {/eq}
• {eq}y = \dfrac{64}{3\times 4} {/eq}
• {eq}\boxed{\color{blue}{y = \dfrac{16}{3}}} {/eq} 