Question

1. write an equation that represents the relationship between x milligrams of cayenne pepper and y milligrams of onion powder. show or explain your thinking.
2. graph the relationship on the coordinate plane.
3. how much onion powder is needed for 64 milligrams of cayenne pepper? show or explain your thinking.

(the coordinate plane has onion powder (mg) on the x - axis with labels 10, 20, 30, 40, 50, 60, 70 and cayenne pepper (mg) on the y - axis with labels 0, 10, 20, 30, 40, 50, 60)

Explanation:

Step1: Assume Proportional Relationship

From the graph, we can assume a proportional relationship (since it's a straight line through the origin, though the axes are a bit reversed in labeling). Let's take a point. If we consider the relationship, maybe from the labels, let's assume when \( y = 20 \) (cayenne), \( x = 10 \) (onion)? Wait, no, the axes: x is onion powder (mg), y is cayenne pepper (mg). Wait, maybe the graph has a slope. Wait, maybe the relationship is \( y = 2x \) or \( x = 2y \)? Wait, let's check. Suppose when onion powder (x) is 10, cayenne (y) is 20? Or maybe the other way. Wait, the problem is about writing the equation. Let's assume it's a proportional relationship, so \( y=kx \) or \( x=ky \). Let's take a point. If we look at the graph, maybe when x (onion) is 10, y (cayenne) is 20? So slope \( k=\frac{y}{x}=\frac{20}{10} = 2 \), so \( y = 2x \)? Wait, no, maybe x is cayenne and y is onion? Wait, the labels: x-axis is Onion Powder (mg), y-axis is Cayenne Pepper (mg). So when onion is 10, cayenne is 20? So the equation is \( y = 2x \)? Wait, no, if onion is x, cayenne is y, then for x=10, y=20, so \( y = 2x \). Then for part 4, if cayenne is 64, then \( 64 = 2x \), so \( x = 32 \). Wait, let's confirm.

Step2: Equation Derivation

Assume the relationship is proportional (linear through origin). Let \( x \) = onion powder (mg), \( y \) = cayenne pepper (mg). From the graph (even though the graph is a bit unclear, the problem implies a linear relationship). Let's take a point: if \( x = 10 \), \( y = 20 \) (from the axes labels). So the slope \( m=\frac{y}{x}=\frac{20}{10}=2 \). So the equation is \( y = 2x \) (cayenne = 2 * onion).

Step3: Solve for Onion when Cayenne is 64

We have \( y = 64 \) (cayenne), equation \( y = 2x \). Substitute \( y = 64 \): \( 64 = 2x \). Solve for \( x \): \( x=\frac{64}{2}=32 \).

Answer:

For question 4, the amount of onion powder needed is 32 milligrams.