Question

lesson 7

1. there is a proportional relationship between the volume of a sample of helium in liters and the mass of that sample in grams. if the mass of a sample is 5 grams, its volume is 28 liters. (5, 28) is shown on the graph below.

(graph: x - axis labeled mass (grams) from 0 to 22, y - axis labeled volume (liters) from 0 to 180, with point (5,28) plotted)
a) what is the constant of proportionality in this relationship?
b) in this situation, what is the meaning of the number you found in part a?
add at least three more points to the graph above, and label with their coordinates.

Explanation:

Part (a)

Step 1: Recall the proportional relationship formula

For a proportional relationship, the formula is \( y = kx \), where \( k \) is the constant of proportionality, \( y \) is the dependent variable (volume), and \( x \) is the independent variable (mass).

Step 2: Substitute the given values

We know that when \( x = 5 \) (mass in grams), \( y = 28 \) (volume in liters). Substitute these values into the formula \( y = kx \):
\( 28 = k \times 5 \)

Step 3: Solve for \( k \)

To find \( k \), divide both sides of the equation by 5:
\( k=\frac{28}{5}=5.6 \)

The constant of proportionality \( k = 5.6 \) in this context represents the ratio of volume to mass for the helium sample. Specifically, it means that for every 1 gram of helium, the volume of the helium sample is 5.6 liters. In other words, the volume of the helium sample (in liters) is 5.6 times the mass of the sample (in grams).

Step 1: Use the proportional relationship \( y = 5.6x \)

We can find additional points by choosing different values of \( x \) (mass) and calculating the corresponding \( y \) (volume) using the formula \( y = 5.6x \).

Step 2: Calculate for \( x = 1 \)

Substitute \( x = 1 \) into \( y = 5.6x \):
\( y = 5.6\times1 = 5.6 \). So the point is \( (1, 5.6) \).

Step 3: Calculate for \( x = 2 \)

Substitute \( x = 2 \) into \( y = 5.6x \):
\( y = 5.6\times2 = 11.2 \). So the point is \( (2, 11.2) \).

Step 4: Calculate for \( x = 10 \)

Substitute \( x = 10 \) into \( y = 5.6x \):
\( y = 5.6\times10 = 56 \). So the point is \( (10, 56) \).

Answer:

The constant of proportionality is \( \boldsymbol{5.6} \) (liters per gram).

Part (b)