Problem 37 - Algebra Form and Function

Question

The cost of denim fabric is directly proportional to the amount that you buy. Let \(C\) be the cost, in dollars, of \(x\) yards of denim fabric. (a) Write a formula expressing \(C\) as a function of \(x\). (b) One type of denim costs \(\$ 28.50\) for 3 yards. Find the constant of proportionality and give its units. (c) How much does 5.5 yards of this type of denim cost?

Step-by-Step Solution

Answer

Answer: The cost of 5.5 yards of this type of denim is $52.25.

Step 1: Writing the formula as a function of x

Since the denim fabric cost is directly proportional to the yardage, we can express the cost (C) as a function of the yardage (x). That would be: $$C=kx$$ where k is the constant of proportionality.

Step 2: Finding the constant of proportionality and its units

The problem states that one type of denim costs \(\$28.50\) for 3 yards. We can substitute these values into our equation from Step 1 and solve for k. $$28.50 = k \cdot 3$$ Divide both sides by 3 to find the value of k: $$k = \frac{28.50}{3} = 9.50$$ So, the constant of proportionality is \(k = \) \(9.50\) dollars per yard. The units for k are dollars per yard since the cost is in dollars and the yardage is in yards.

Step 3: Finding the cost of 5.5 yards of denim

Now, we need to find the cost of 5.5 yards of this type of denim. We can use the formula from Step 1, including the value of k from Step 2, to find the cost. $$C = 9.50 \cdot x$$ Substitute the value of x (5.5 yards) into the equation: $$C = 9.50 \cdot 5.5$$ Multiply the constant by the yardage to get the cost: $$C = 52.25$$ So, 5.5 yards of this type of denim costs \(\$52.25\).

Key Concepts

Constant of Proportionality

When two quantities have a direct proportional relationship, their ratio remains constant. In the context of our denim example, the cost of denim changes at a constant rate per yard. This rate is known as the constant of proportionality, denoted by "k" in our equation.
Let's delve deeper:

• This constant makes it easy to predict one quantity by knowing the other.
• To find "k," you divide the total cost by the number of yards.
• In our example, if 3 yards of denim costs \(28.50, then:

\[ k = \frac{C}{x} = \frac{28.50}{3} = 9.50 \]
This tells us each yard costs \)9.50. So, for every yard we increase, the cost increases by $9.50. Hence, the constant of proportionality helps translate relationships into useful predictions.

Linear Functions

Linear functions are a fundamental concept in mathematics and describe relationships with a constant rate of change. These functions are often depicted as straight lines on a graph. In our denim scenario, the cost function is linear.
Here's why:

• The equation \( C = kx \) is a linear function since it’s of the form \( y = mx + b \) with \( b = 0 \).
• The coefficient "k" represents the slope of the line, indicating how much \( C \) changes as \( x \) changes.
• When plotted, the curve illustrating cost vs. fabric length will be a straight line.

With linear functions, the relationship remains straightforward and consistent. If you double the amount of denim, the cost doubles. Understanding linear functions provide insights into predictable and consistent relationships.

Unit Rate

The unit rate is an essential component when discussing direct proportionality, especially in practical applications like shopping. It represents the cost or value per single unit of measure.
Thinking about our fabric example:

• The unit rate is the cost per yard, which we've found to be $9.50 per yard using the constant of proportionality.
• Finding the unit rate helps us quickly determine the cost of any amount of fabric without recalculating from the total price data each time.
• This simplifies budgeting and buying decisions because you can multiply the unit rate by any number of yards you wish to purchase to get the total cost.

Knowing the unit rate is powerful. It transforms complex problems into simple arithmetic, beneficial for consumers and sellers in pricing and buying strategies.