Partial Variation: Understanding Cost Relationships in Real-World Examples

Partial Variation

Example 1: A banquet hall charges a flat fee of $4000 for an event plus a charge of $100 per person attending. Write a mathematical model to calculate the cost of an event. Does the cost of an event depend on the number of people that attend the event?
Let C represent the cost of renting the hall and let p represent the number of people attending the banquet.
C=100p+4000
Costs depend partially on the number of people attending the party.

(Partial variation)
A partial variation has an equation of the form y=mx+b, where m and b are constants, and
• m is the constant of variation
• b is the initial value (fixed value)
Note that when x = 0, y = initial value (not zero). The graph of a partial variation does not pass through the origin.
Example 2: Represent the following relationship using a mathematical model. The cost of a gym membership is $100 plus $30 per month.
a) Identify your variables.

Use Let statements.
Let C represent the cost of the gym membership and let ‘n represent the number of mouths
b) Which variable is the independent? dependent?
n is the independent variable and C is the dependent variable.
c) Create a table of values, include at least 4 values

d) Graph the relationship. Include appropriate labels.

e) What is the rate of change? What does it represent?
m=Change in costChange in the number of months
=301
=30
So, the rate of change is 30.
The rate of change (constant of variation) represents the cost of the gym membership per month.

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f) What is the initial value? What does it represent?
The initial value (fixed cost) is 100. It represents the initial cost (even if you only signed) up but didn`t go to the gym, you still pay $100.
b=100
g) Write an equation to represent this relationship.
y=mx+b
C=30n+100

Example 3: The following tables represent the relationship between the cost, in dollars, and the number of videos rented. Identify the relation as being partial or direct. Justify your answer.

Direct (passes through the origin)
m=51
=5

Variation: 5
Partial since there is an initial cost (fee)
m=21
=2
Variation: 2