Question

1. the table shows the amount of dietary fiber in bananas. which of the following statements are true? select all that apply.

a. the constant of proportionality is 3.1
b. a graph of the table would be a straight line throught the origin.
c. the ordered pair (3.1, 1) would represent the unit rate.
d. the ordered pair (6, 18.6) means 6 grams of fiber in 18.6 bananas.
e. the ordered pair (15, 46.5) would be a point on this graph.

Explanation:

Step1: Calculate the constant of proportionality

For a proportional relationship \(y = kx\), where \(y\) is fiber (\(g\)), \(x\) is bananas.
Take the first pair \((x = 3,y=9.3)\), \(k=\frac{y}{x}=\frac{9.3}{3}=3.1\).
Check other pairs: \(\frac{18.6}{6}=3.1\), \(\frac{27.9}{9}=3.1\), \(\frac{37.2}{12}=3.1\). So the constant of proportionality \(k = 3.1\).

Step2: Analyze the graph of a proportional relationship

The equation of the relationship is \(y = 3.1x\). The graph of \(y=mx\) (\(m
eq0\)) is a straight - line through the origin \((0,0)\) since when \(x = 0\), \(y=3.1\times0 = 0\).

Step3: Analyze the unit - rate ordered pair

The unit rate is when \(x = 1\). Substitute \(x = 1\) into \(y=3.1x\), we get \(y = 3.1\). So the ordered pair \((1,3.1)\) represents the unit rate. The ordered pair \((3.1,1)\) is incorrect.

Step4: Analyze the meaning of the ordered pair \((6,18.6)\)

In the context of the table, the first value in the ordered pair is the number of bananas and the second is the amount of fiber. So \((6,18.6)\) means \(18.6\) grams of fiber in \(6\) bananas, not the other way around.

Step5: Analyze the ordered pair \((15,46.5)\)

Substitute \(x = 15\) into \(y = 3.1x\). \(y=3.1\times15=(3 + 0.1)\times15=45+1.5 = 46.5\). So \((15,46.5)\) is a point on the graph \(y = 3.1x\).

Answer:

A. The constant of proportionality is 3.1; B. A graph of the table would be a straight line throught the origin; E. The ordered pair \((15,46.5)\) would be a point on this graph.