Question

a dishwasher uses the same number gallons of water for each load. the table shows how many gallons of water, g, the dishwasher uses for n loads of dishes. what equation can you use to represent the relationship shown in the table? show your work.

dishwasher water use
number of loads, n | gallons used, g
4 | 18.0
5 | 22.5
6 | 27.0
7 | 31.5

Explanation:

Step1: Find the rate of change

To find the relationship between the number of loads \( n \) and gallons of water \( g \), we first calculate the rate of change (slope). Let's take two points, say \( (n_1, g_1)=(4, 18.0) \) and \( (n_2, g_2)=(5, 22.5) \). The slope \( m \) is given by \( m=\frac{g_2 - g_1}{n_2 - n_1} \).
\[
m=\frac{22.5 - 18.0}{5 - 4}=\frac{4.5}{1}=4.5
\]
We can check with another pair, like \( (5, 22.5) \) and \( (6, 27.0) \):
\[
m=\frac{27.0 - 22.5}{6 - 5}=\frac{4.5}{1}=4.5
\]
And \( (6, 27.0) \) and \( (7, 31.5) \):
\[
m=\frac{31.5 - 27.0}{7 - 6}=\frac{4.5}{1}=4.5
\]
So the rate of change (slope) is \( 4.5 \) gallons per load.

Step2: Determine the equation

Since this is a linear relationship, we can use the slope - intercept form of a linear equation \( g = mn + b \). We know the slope \( m = 4.5 \), and we can find the y - intercept \( b \) by plugging in one of the points. Let's use \( n = 4 \) and \( g = 18.0 \):
\[
18.0=4.5\times4 + b
\]
\[
18.0 = 18.0+b
\]
Subtract \( 18.0 \) from both sides: \( b=18.0 - 18.0 = 0 \)
So the equation that represents the relationship is \( g = 4.5n \)

Answer:

The equation is \( g = 4.5n \)