Question

decide whether or not each equation represents a proportional relationship.
a. volume measured in cups (c) vs. the same volume measured in ounces (z): $c = \frac{1}{8}z$
b. area of a square (a) vs. the side length of the square (s): $a = s^2$
c. perimeter of an equilateral triangle (p) vs. the side length of the triangle (s): $3s = p$
d. length (l) vs. width (w) for a rectangle whose area is 60 square units: $l = \frac{60}{w}$

Explanation:

Step1: Recall Proportional Relationship

A proportional relationship is of the form \( y = kx \) (direct proportion) or \( y=\frac{k}{x} \) is inverse proportion, where \( k \) is a constant and the relationship passes through the origin (\( x = 0,y = 0 \) for direct). For direct proportion, the ratio \( \frac{y}{x}=k \) (constant).

Step2: Analyze Part (a)

Equation: \( c=\frac{1}{8}z \). This is in the form \( c = kz \) with \( k=\frac{1}{8} \) (constant). When \( z = 0 \), \( c = 0 \). So it is a proportional (direct) relationship.

Step3: Analyze Part (b)

Equation: \( A=s^{2} \). The ratio \( \frac{A}{s}=s \), which is not a constant (depends on \( s \)). So it is not a proportional relationship.

Step4: Analyze Part (c)

Equation: \( 3s = P \) or \( P = 3s \). This is in the form \( P=ks \) with \( k = 3 \) (constant). When \( s = 0 \), \( P = 0 \). So it is a proportional (direct) relationship.

Step5: Analyze Part (d)

Equation: \( L=\frac{60}{w} \) or \( Lw=60 \). The ratio \( \frac{L}{w}=\frac{60}{w^{2}} \), which is not a constant (depends on \( w \)). Also, it is an inverse relationship but for proportional (direct) we need \( \frac{y}{x}=\text{constant} \), here it's not. So it is not a proportional relationship.

Answer:

a. Proportional Relationship
b. Not a Proportional Relationship
c. Proportional Relationship
d. Not a Proportional Relationship