When the owner of a gas station sets the price of 1 gallon of unleaded gasoline at $2, he can sell approximately 15 comma 000 gallons per day. When he sets the price at $5 per gallon, he can sell approximately 9 comma 000 gallons. Let G(x) denote the number of gallons of unleaded gasoline sold 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 is set at $2.50 per gallon

14.75  gallons per day

Explanation:

To solve this question, we make use of the linear supply function equation of the following form:

Q(x) = a + bx   (1)

Where

Q = number of gallons of unleaded gasoline sold per day  at x dollars

a = number of gallons of unleaded gasoline sold that is independent of price

b = the coefficient of price x

x = the price of gallons of unleaded gasoline

At G($2) = 15, equation (1) becomes:

15 = a + b2   (2)

At G($5) = 9, equation (1) becomes:

9 = a + b*5   (3)

To get a and b, we make a the subject of the formula in equation (2) and then substitute into equation three as follows:

a = 15 - 2b (4)

Substituting for a in equation (4) into equation (3), we have:

9 = (15 - 2b) + 5b

9 = 15 - 2b + 5b

By rearranging, we have:

9 - 15 = 5b - 2b

-6 = 3b

b = 3/(-6)

b = - 0.5 (5)

To get the value of a, substitute -0.5 for b in equation (4) as follows:

a = 15 - 2(-0.5)

a = 15 + 1

a = 16 (6)

Substituting for b and a from equations (5) and (6) into equation (1), we will derive the linear supply function for this question as follows:

Q(x) = a + bx  

Q(x) = 16 - 0.5x (7)

The number of gallons that will be sold per day if the price is set at $2.50 per gallon can be calculated by substituting $2.50 for x in equation (7) as follows:

Q($2.50) = 16 - 0.5(2.50)

               = 16 - 1.25

               = 14.75  gallons per day

Therefore, 14.75 comma 000  gallons per day will sold  if the price is set at $2.50 per gallon.

Note

Since 15 comma 000 gallons per day can be sold at the price of $2, 14.75 comma 000 gallons per day can be sold at the price of $2.5, and 9 comma 000 gallons per day can be sold at the price of $5, this shows that there is a negative relationship between the price and the number of gallon sold per day. That is, as price increases, the number of quantity sold decreases. That is why equation (7) obtained above is a negative linear supply equation due to that -0.5x

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