Question

the table represents the amount of coffee in a coffee pot based on the number of minutes the coffee has been brewing.

time (min)	coffee (oz)
3	7.2
4	9.6
5	12

____ 4. the ratio of \\(\frac{y}{x}\\) is not constant.

____ 5. the table represents a proportional relationship between x and y.

____ 6. the table can be represented by \\(y = x + 2.4\\).

two students wrote the equations shown below.

erica: \\(y = -0.5x\\)

aliyah: \\(y = 2.5x - 8\\)

Explanation:

Step1: Analyze Question 4

To check if the ratio \(\frac{y}{x}\) is constant, we calculate \(\frac{y}{x}\) for each pair \((x,y)\) from the table.

• For \(x = 2\), \(y = 4.8\): \(\frac{4.8}{2}=2.4\)
• For \(x = 3\), \(y = 7.2\): \(\frac{7.2}{3}=2.4\)
• For \(x = 4\), \(y = 9.6\): \(\frac{9.6}{4}=2.4\)
• For \(x = 5\), \(y = 12\): \(\frac{12}{5}=2.4\)

The ratio \(\frac{y}{x}\) is constant (2.4), so the statement "The ratio of \(\frac{y}{x}\) is not constant" is False. The correct statement is "The ratio of \(\frac{y}{x}\) is constant".

Step2: Analyze Question 5

A proportional relationship has the form \(y = kx\) (where \(k\) is constant), and we saw \(\frac{y}{x}=k = 2.4\) (constant). So the table represents a proportional relationship between \(x\) and \(y\). Thus, the statement is True.

Step3: Analyze Question 6

We check if \(y=x + 2.4\) fits the table.

• For \(x = 2\): \(y=2 + 2.4=4.4

eq4.8\)

• For \(x = 3\): \(y=3 + 2.4=5.4

eq7.2\)
The equation \(y=x + 2.4\) does not fit the table. The correct equation is \(y = 2.4x\) (since \(\frac{y}{x}=2.4\)). So the statement "The table can be represented by \(y=x + 2.4\)" is False. The correct equation is \(y = 2.4x\).

Answer:

1. False; The ratio of \(\frac{y}{x}\) is constant.
2. True
3. False; The table can be represented by \(y = 2.4x\).