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.

(graph: x - axis labeled onion powder (mg) with 0,10,20,30,40,50,60,70; y - axis labeled cayenne pepper (mg) with 0,10,20,30,40,50,60)

Explanation:

Step1: Assume Proportional Relationship

Assume \( y = kx \) (proportional, since graph likely linear through origin). Need to find \( k \). But maybe from typical ratios, or if we assume when \( x = 30 \), \( y = 20 \)? Wait, maybe the graph has points. Wait, the grid: let's check coordinates. Suppose when \( x = 30 \) (onion), \( y = 20 \) (cayenne)? Wait, no, maybe inverse? Wait, the problem 4 is: 64 mg cayenne, find onion. Let's assume a proportional relationship (direct variation) \( y = kx \), but maybe \( x \) (onion) and \( y \) (cayenne) have a ratio. Wait, maybe from the graph, when \( x = 30 \), \( y = 20 \)? Wait, no, let's re-examine. Wait, the x-axis is Onion Powder (mg), y-axis Cayenne Pepper (mg). Let's suppose the line passes through (30,20)? Wait, no, maybe (20,30)? Wait, maybe I misread. Wait, problem 4: 64 mg cayenne, find onion. Let's assume \( \frac{x}{y} = \frac{30}{20} = \frac{3}{2} \) (onion to cayenne). So \( x = \frac{3}{2}y \). If \( y = 64 \), then \( x = \frac{3}{2} \times 64 = 96 \)? Wait, no, maybe inverse. Wait, maybe \( y = \frac{2}{3}x \). Wait, let's think again. Suppose when onion is 30 mg, cayenne is 20 mg. So ratio onion:cayenne = 3:2. So for 64 mg cayenne, onion = \( 64 \times \frac{3}{2} = 96 \) mg. Wait, but let's check the graph. The x-axis (onion) has 0,10,20,30,40,... y-axis (cayenne) 0,10,20,30,40,... So if the line goes through (30,20), then slope \( k = \frac{20}{30} = \frac{2}{3} \), so \( y = \frac{2}{3}x \). Wait, no, \( y \) is cayenne, \( x \) onion. So \( y = \frac{2}{3}x \) would mean when \( x = 30 \), \( y = 20 \), which matches. So to find \( x \) when \( y = 64 \): \( 64 = \frac{2}{3}x \) → \( x = 64 \times \frac{3}{2} = 96 \).

Step2: Calculate Onion Powder

Using \( y = \frac{2}{3}x \), solve for \( x \) when \( y = 64 \):
\( x = \frac{3}{2}y = \frac{3}{2} \times 64 = 96 \).

Answer:

96 milligrams