Question

A daycare facility charges a fee for every minute that a parent is late after the facility’s published closing time. The table represents the fee, y, which is dependent on the number of minutes late, x.

x
y
5
32
8
50
11
68

Which of the following linear equations represents the situation?
y = 3 x + 22
y = 6 x + 36
y = 6 x + 2
y = 24 x + 36

Answer 1

To determine which linear equation represents the relationship between the number of minutes late, \(x\), and the fee charged, \(y\), we first need to identify the slope and y-intercept based on the provided data.

From the table, we have the following points:
- When \(x = 5\), \(y = 32\)
- When \(x = 8\), \(y = 50\)
- When \(x = 11\), \(y = 68\)

First, we can calculate the slope \(m\) using two points. Let's use the first two points \((5, 32)\) and \((8, 50)\):

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{50 - 32}{8 - 5} = \frac{18}{3} = 6
\]

Now that we have the slope \(m = 6\), we can use one of the points to find the y-intercept \(b\).

Using the point \((5, 32)\):

\[
y = mx + b \Rightarrow 32 = 6(5) + b
\]
\[
32 = 30 + b \Rightarrow b = 32 - 30 = 2
\]

Thus, the linear equation that represents the situation is:

\[
y = 6x + 2
\]

Checking this with another point \((8, 50)\) to confirm:
\[
y = 6(8) + 2 = 48 + 2 = 50
\]

This equation is consistent with the data. Therefore, the correct linear equation that represents the situation is:

\[
\boxed{y = 6x + 2}
\]