Question

the relationship between x and y can be modeled by an inverse variation function $y = \frac{k}{x}$. \

flow rate, x (gal/hr)	time, y (hr)	\
----	----	\
300	66	\
400	49.5	\
500	39.6	\
600	33	\

what is the value of k for the inverse variation function?\
$k = \boldsymbol{\text{dropdown}}$\
what does k represent in this situation?\
$\boldsymbol{\text{dropdown}}$\
which rational function represents this situation?\
$\boldsymbol{\text{dropdown}}$

Explanation:

Step1: Find k using first row

Given \( y = \frac{k}{x} \), substitute \( x = 300 \), \( y = 66 \).
\( 66 = \frac{k}{300} \) → \( k = 66 \times 300 = 19800 \).
Check with other rows (e.g., \( x = 400 \), \( y = 49.5 \)): \( 49.5 \times 400 = 19800 \), consistent.

Step2: Interpret k

In flow rate (\( x \), gal/hr) and time (\( y \), hr), \( k = x \times y \) is volume (gal), since \( \text{Volume} = \text{Flow Rate} \times \text{Time} \).

Step3: Rational function

From \( y = \frac{k}{x} \) and \( k = 19800 \), the function is \( y = \frac{19800}{x} \).

Answer:

1. \( k = 19800 \)
2. \( k \) represents the total volume of the liquid (in gallons)
3. The rational function is \( y = \frac{19800}{x} \)