Question

writing equations using constant of proportionality
cake recipes

1. the cake machine needs to be programmed to make cakes. the recipe says to ounces of milk for every 3 cakes.

number of cakes (c) | ounces of milk (m)
3 | 18
6 | 36
12 | 72
a. complete the table for different batches o
b. what is the constant of proportionality?
c. what does that constant of proportionality te you about this situation?
cake recipes (continued)
d. write an equation to model how many ounces of milk, m, are needed for c, cakes.

Explanation:

Part a: Completing the Table

Step 1: Find the constant of proportionality (pre - step for table completion)

We know that for proportional relationships, \(m = kc\), where \(k\) is the constant of proportionality. From the first row, when \(c = 3\), let's assume the correct value of \(m\) (the hand - written 18 seems more appropriate for a proportional relationship, maybe a writing error in 19). If \(c = 3\) and \(m = 18\), then \(k=\frac{m}{c}=\frac{18}{3}=6\).

Step 2: Complete the table

• For \(c = 3\), using \(k = 6\), \(m=k\times c=6\times3 = 18\).
• For \(c = 6\), \(m=k\times c=6\times6 = 36\) (which matches the given value).
• For \(m = 72\), we solve \(72=k\times c\). Since \(k = 6\), then \(c=\frac{72}{6}=12\).

So the completed table (correcting the first \(m\) value) is:

Number of Cakes (\(c\))	Ounces of Milk (\(m\))
6	36
12	72

Part b: Constant of Proportionality

Step 1: Recall the formula for constant of proportionality

For a proportional relationship between two variables \(m\) (ounces of milk) and \(c\) (number of cakes), the constant of proportionality \(k\) is given by \(k=\frac{m}{c}\).

Step 2: Calculate \(k\)

Using the values from the table, when \(c = 3\) and \(m = 18\), \(k=\frac{m}{c}=\frac{18}{3}=6\).

Part c: Interpretation of the Constant of Proportionality

The constant of proportionality \(k = 6\) means that for this cake - making process, each cake requires 6 ounces of milk. In other words, the ratio of the number of ounces of milk to the number of cakes is always 6:1. So, for every 1 cake that the machine makes, it needs 6 ounces of milk.

Part d: Equation for the Relationship

Step 1: Recall the form of a proportional equation

A proportional relationship between two variables \(m\) (dependent variable) and \(c\) (independent variable) is of the form \(m=kc\), where \(k\) is the constant of proportionality.

Step 2: Substitute the value of \(k\)

We found that \(k = 6\), so the equation that models the number of ounces of milk \(m\) needed for \(c\) cakes is \(m = 6c\).

Answer:

s:
a. The completed table is as shown above.
b. The constant of proportionality is \(\boldsymbol{6}\).
c. It means each cake requires 6 ounces of milk.
d. The equation is \(\boldsymbol{m = 6c}\).