The total cost of gasoline varies directly with the number of gallons purchased. Gas costs $1.77 per gallon. Write a direct variation to model the total cost [tex]$c$[/tex] for [tex]$g$[/tex] gallons of gas.
[tex]$g=1.77 c$[/tex]
[tex]$c=1.77 g$[/tex]
[tex]$c=\frac{8}{177}g$[/tex]
Answer (2)
The total cost c varies directly with the number of gallons g .
The cost per gallon is $1.77, which is the constant of variation.
The direct variation equation is c = 1.77 g .
Therefore, the equation that models the total cost is c = 1.77 g .
Explanation
Understanding the Problem Let's analyze the problem. We are given that the total cost of gasoline varies directly with the number of gallons purchased. This means that there is a constant price per gallon. We are given that gas costs 1.77 p er g a ll o n . W e n ee d t o w r i t e a d i rec t v a r ia t i o n e q u a t i o n t o m o d e lt h e t o t a l cos t c f or g$ gallons of gas.
Setting up the Equation Since the total cost c varies directly with the number of gallons g , we can write the equation as: c = k g where k is the constant of variation, which represents the cost per gallon.
Finding the Constant of Variation We are given that the cost per gallon is 1.77 , so w es u b s t i t u t e k = 1.77 in t o t h ee q u a t i o n : c = 1.77 g $
Final Equation Therefore, the direct variation equation that models the total cost c for g gallons of gas is c = 1.77 g .
Examples
Direct variation is a concept that applies to many real-life situations. For example, the amount you earn at a job where you are paid hourly varies directly with the number of hours you work. If you earn $15 per hour, your total earnings E can be modeled as E = 15 h , where h is the number of hours you work. Similarly, the distance you travel at a constant speed varies directly with the time you travel. If you drive at 60 miles per hour, the distance d you travel can be modeled as d = 60 t , where t is the time in hours. Understanding direct variation helps you predict outcomes based on constant rates or ratios.
The total cost of gasoline varies directly with the number of gallons purchased. The relationship can be expressed with the equation c = 1.77 g , where c is the total cost and g is the number of gallons. This equation indicates that for every gallon purchased at $1.77 each, the total cost increases proportionately.
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