Question

7th grade math ia 3, l068 (proportional relationships)
constant of proportionality - graphs
homework
a cereal company is developing a new granola bar. it follows a recipe based on the graph shown below.

1. what is the constant of proportionality?
2. explain what the constant of proportionality means for this example. your response should mention

uts\ and \fruit\.

1. how many cups of nuts would be needed for 10 cups of fruit?

show or explain how you know.

1. how many cups of fruit would be needed for 9 cups of nuts?

show or explain how you know.

1. make a table of the graph above. include at least five pairs of values.

(graph: x-axis labeled nuts (cups) from 0 to 8, y-axis labeled fruit (cups) from 0 to 8, with a line passing through points (1,2), (2,4), (3,6), (4,8))

Explanation:

Question 1:

Step1: Recall the formula for constant of proportionality

For a proportional relationship \( y = kx \), the constant of proportionality \( k=\frac{y}{x} \), where \( x \) is the number of cups of nuts and \( y \) is the number of cups of fruit.

Step2: Pick a point from the graph

From the graph, when \( x = 1 \) (nuts), \( y = 2 \) (fruit).

Step3: Calculate \( k \)

Using \( k=\frac{y}{x} \), substitute \( x = 1 \) and \( y = 2 \). So \( k=\frac{2}{1}=2 \).

The constant of proportionality \( k = 2 \) means that for every 1 cup of nuts, the recipe uses 2 cups of fruit. In other words, the amount of fruit (in cups) is 2 times the amount of nuts (in cups) used in the granola bar recipe.

Step1: Use the proportional relationship

The relationship is \( y = 2x \) (since \( k = 2 \), \( y \) is fruit, \( x \) is nuts). We know \( y = 10 \) (fruit), solve for \( x \) (nuts).

Step2: Substitute \( y = 10 \) into \( y = 2x \)

\( 10 = 2x \)

Step3: Solve for \( x \)

Divide both sides by 2: \( x=\frac{10}{2}=5 \).

Answer:

The constant of proportionality is \( 2 \).

Question 2: