Problem 33 - Algebra Form and Function

Question

The total cost of purchasing gasoline for your car is directly proportional to the the number of gallons pumped, and 11 gallons cost \(\$ 36.63\). (a) What is the constant of proportionality, and what are its units? (b) How much do 15 gallons cost?

Step-by-Step Solution

Answer

Answer: (a) The constant of proportionality is $3.33 per gallon. (b) The cost for 15 gallons is $49.95.

Step 1: Understand the direct variation formula

We are given that the total cost (C) is directly proportional to the number of gallons pumped (G). Therefore, we can use the direct variation formula: C = kG where 'k' is the constant of proportionality.

Step 2: Find the constant of proportionality and units

We are given that 11 gallons cost $36.63. We can plug this information into the direct variation formula to find the constant of proportionality 'k': \$36.63 = k * 11 To find k, we will divide both sides by 11: k = \$36.63 / 11 k = \$3.33 So, the constant of proportionality (k) is \$3.33 per gallon. The units are dollars per gallon.

Step 3: Find the cost for 15 gallons

Now we know k = \$3.33, we can use the direct variation formula to find the cost of 15 gallons: C = kG C = \$3.33 * 15 C = \$49.95 So, the cost of 15 gallons is \$49.95.

Key Concepts

Constant of Proportionality

In the context of direct variation, the constant of proportionality represents the fixed rate at which one variable changes with another. Specifically, in our gasoline cost problem, it tells us how much each gallon of gasoline costs. In mathematical terms, if the total cost \( C \) is directly proportional to the number of gallons \( G \), then:

• \( C = kG \)

Here, \( k \) is the constant of proportionality.
The constant is calculated by dividing the total cost by the number of gallons. For this example, using the given data:

• 11 gallons cost \( \\(36.63 \)

Using the formula, we find \( k \) by dividing 36.63 by 11:

• \( k = \\)36.63 / 11 = \\(3.33 \)

This means each gallon costs \\)3.33. Thus, the constant of proportionality, \( k \), has units of dollars per gallon, indicating how each unit change in gallons affects the total cost.

Unit Conversion

Understanding units is crucial as they tell us how to interpret our numbers. In this exercise, we are dealing with dollars and gallons. The constant of proportionality \( k \) has the units of dollars per gallon, highlighted by its calculation.
When given a numeric answer like \( k = \\(3.33 \), it's essential to include its units, which indicate what each piece of data represents in a real-world scenario. In our equation \( C = kG \),

• \( k \) is in \( \\)/\text{gal} \),
• \( G \) is in \( \text{gallons} \).

Therefore, correctly understanding unit conversions aids in knowing how changes in gallons affect costs directly. Though no conversion from one unit to another (like from gallons to liters or inches to centimeters) is needed here, clearly defining units is foundational in identifying how units interact in calculations.

Rate Calculation

Rate calculation in this context involves determining how much something happens per unit of another measurement. For our problem, the rate is the cost per gallon of gasoline.
This rate gives us the ability to calculate the total cost for any number of gallons or alter other variables, making it extremely useful.

• Once the rate, or the constant of proportionality \( k \), is known: \( k = \\(3.33/\text{gal} \),

The rate allows us to make quick calculations of total cost by multiplying it by the number of gallons, exemplified in the solution when calculating the cost of 15 gallons:

• \( C = k \times G = \\)3.33 \times 15 = \$49.95 \)

Understanding this type of rate calculation helps in many practical scenarios, like budgeting or price comparison, because it straightforwardly connects changes in the input directly to changes in the output.