When the owner of a gas station sets the price of 1 gallon of unleaded gasoline at 2$, he can seek approximately 20,000 gallons per day. When he sets the price of gas at 5$ per gallon, he can sell approximately 14,000 gallons per day. Let G(x) denote the number of gallons of unleaded gasoline per day when the price is x dollars. Assume that G(x) is a linear function of x. Approximately how many gallons will be sold per day if the price of gasoline is 2.50$ per gallon?
To find the approximate number of gallons that will be sold per day when the price of gasoline is $2.50 per gallon, we can use linear interpolation.
Given the information provided, we have two data points:
- When the price is $2, the owner can sell approximately 20,000 gallons per day.
- When the price is $5, the owner can sell approximately 14,000 gallons per day.
We need to find the number of gallons that will be sold when the price is $2.50.
To calculate this, we can use the concept of a linear function:
G(x) = mx + b
where G(x) is the number of gallons sold, x is the price of gasoline, m is the slope, and b is the y-intercept.
First, we need to find the slope (m) of the linear function:
m = (change in G) / (change in x)
m = (14,000 - 20,000) / (5 - 2)
m = -6,000 / 3
m = -2000
Next, we can find the y-intercept (b) by substituting one of the data points into the equation:
20,000 = (-2000)(2) + b
20,000 = -4000 + b
b = 20,000 + 4000
b = 24,000
Now that we have the slope and y-intercept, we can substitute in the price $2.50 to find the number of gallons sold (G(x)):
G(2.50) = (-2000)(2.50) + 24,000
G(2.50) = -5000 + 24,000
G(2.50) = 19,000
Therefore, approximately 19,000 gallons will be sold per day if the price of gasoline is $2.50 per gallon.