A manufacture has been selling 1300 television sets a week at 450 each. A market survey indicates that for each 14 rebate offered to a buyer, the number of sets sold will increase by 140 per week.
a) Find the demand function , where is the number of the television sets sold per week.
Demand function: D(r) = 1300 + 140r, where r is the rebate offered.
To find the demand function for the television sets, we need to consider the relationship between the price of the television sets and the quantity demanded.
Given that the manufacturer has been selling 1300 television sets a week at $450 each, we can assume that at that price, there is a certain level of demand. Let's call this demand level Q0.
We also know that for each $14 rebate offered to a buyer, the number of sets sold will increase by 140 per week. This indicates that the quantity demanded is directly impacted by the rebate amount.
Let's denote the rebate amount as R and the increase in quantity demanded as D. From the given information, we know that D = 140 and R = $14.
Using this information, we can write the demand function as:
Q = Q0 + DP
Here, Q is the quantity demanded, Q0 is the demand level at the original price, D is the increase in quantity demanded per increase in rebate amount, and P is the price.
To find the demand function, we need to determine the demand level at the original price, Q0. We know that at a price of $450, the quantity demanded is 1300 sets per week. So, we can substitute this into the equation:
1300 = Q0 + (140 * 0)
1300 = Q0
Therefore, we can now write the demand function as:
Q = 1300 + 140P
This is the demand function for the television sets, where Q is the number of sets sold per week and P is the price.
To find the demand function, we need to determine the relationship between the number of sets sold per week and the price of the sets, as well as the impact of a rebate.
Let's denote the number of sets sold per week as x and the price of each set as p.
Given that the manufacturer has been selling 1300 sets per week at $450 each, we can write the initial demand equation as:
1300 = x * 450
Next, the market survey indicates that for each $14 rebate offered to a buyer, the number of sets sold will increase by 140 per week. This implies that the price per set decreases by $14.
We can represent the relationship between the price (p) and the number of sets sold (x) as follows:
p = 450 - 14 * (x - 1300/140)
Simplifying the equation, we get:
p = 450 - 14 * (x - 9.2857)
p = 450 - 14x + 129.9998
p = -14x + 579.9998
Rearranging the equation to express x in terms of p:
14x = -p + 579.9998
x = (-p + 579.9998) / 14
Therefore, the demand function is given by:
x = (-p + 579.9998) / 14