Question

suppose that y is inversely proportional to x. find the constant of proportionality k if y = 5 when x = 8. k = \boxed{} using the k from above write the variation equation in terms of x. y = \boxed{} using the k from above find y given that x = 31. y = \boxed{} if needed, round answer to 3 decimal places. enter dne for does not exist, oo for infinity

Explanation:

Step1: Recall inverse proportionality formula

If \( y \) is inversely proportional to \( x \), the formula is \( y=\frac{k}{x} \), where \( k \) is the constant of proportionality.

Step2: Find \( k \) when \( y = 5 \) and \( x = 8 \)

Substitute \( y = 5 \) and \( x = 8 \) into \( y=\frac{k}{x} \). We get \( 5=\frac{k}{8} \). Multiply both sides by 8: \( k = 5\times8=40 \).

Step3: Write the variation equation

Using \( k = 40 \) in \( y=\frac{k}{x} \), the equation is \( y=\frac{40}{x} \).

Step4: Find \( y \) when \( x = 31 \)

Substitute \( x = 31 \) into \( y=\frac{40}{x} \). So \( y=\frac{40}{31}\approx1.290 \) (rounded to 3 decimal places).

Answer:

For \( k \): \( 40 \)
For the variation equation: \( \frac{40}{x} \)
For \( y \) when \( x = 31 \): \( \approx1.290 \)