Question

study how the example shows a proportional relationship. then solve problems 1 - 5. example the double number line shows the pounds of clay jamal uses to make different numbers of identical bowls. is there a proportional relationship between the pounds of clay and the number of bowls? if so, what is the constant of proportionality for pounds of clay per bowl? all of the ratios for clay : bowls are equivalent. so, there is a proportional relationship. all of the ratios have the same rate, \\( \frac { 2 } { 3 } \\) pound of clay per bowl. the constant of proportionality is \\( \frac { 2 } { 3 } \\). 1 a. look at the example. write an equation that shows the relationship between the pounds of clay jamal uses, c, and the number of bowls he makes, b. b. explain how to find the pounds of clay jamal needs to make 8 bowls. 2 zara has paper rectangles of different sizes. every rectangle is 5 cm longer than it is wide. is there a proportional relationship between the lengths and widths of these rectangles? explain.

Explanation:

Step1: Determine the relationship type

Since there is a proportional relationship with a constant of proportionality \(k = \frac{2}{3}\), the general form of a proportional equation is \(y=kx\). Here, \(c\) (clay) is like \(y\) and \(b\) (bowls) is like \(x\).

Step2: Write the equation

Substitute \(k=\frac{2}{3}\), \(y = c\), and \(x = b\) into \(y = kx\). The equation is \(c=\frac{2}{3}b\).

Step3: Explain finding clay for 8 bowls

For part b, use the equation \(c=\frac{2}{3}b\). Substitute \(b = 8\) into the equation. So \(c=\frac{2}{3}\times8=\frac{16}{3}=5\frac{1}{3}\) pounds. We use the proportional equation because the relationship between clay and bowls is proportional (constant ratio of \(\frac{2}{3}\) pound per bowl)

Step4: Analyze rectangle relationship

Let the width of the rectangle be \(w\) and the length be \(l\). Given \(l=w + 5\). For a proportional relationship, \(l=kw\) (where \(k\) is a non - zero constant). If we check the ratios \(\frac{l}{w}=\frac{w + 5}{w}=1+\frac{5}{w}\), which is not a constant (changes as \(w\) changes). For example, if \(w = 1\), \(l=6\) and \(\frac{l}{w}=6\); if \(w = 2\), \(l = 7\) and \(\frac{l}{w}=3.5\)

Answer:

a. \(c=\frac{2}{3}b\)
b. Substitute \(b = 8\) into the equation \(c=\frac{2}{3}b\). Then \(c=\frac{2}{3}\times8=\frac{16}{3}=5\frac{1}{3}\) pounds. We use the equation because of the proportional relationship (constant of proportionality \(\frac{2}{3}\))

1. No. Let \(w\) be the width and \(l\) be the length (\(l=w + 5\)). The ratio \(\frac{l}{w}=\frac{w + 5}{w}=1+\frac{5}{w}\) is not constant. For example, when \(w = 1\), \(\frac{l}{w}=6\); when \(w=2\), \(\frac{l}{w}=3.5\)