A manufacture has been selling 1400 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 function representing the demand , where is the number of the television sets sold per week and is the corresponding price.
b) How large rebate should the company offer to a buyer, in order to maximize its revenue?
c) If the weekly cost function is , how should it set the size of the rebate to maximize its profit?
a) appears to contain some typos
I will do it my way
let the number of $14 rebates be
Return = number sold x price of each unit
= (450 - 14n)(1400 + 140n)
dR/dn = (450-14n)(140) + (1400+140n)(-14)
= 63000 - 1960n - 19600 - 1960n
= 0 for a max of Revenue
3920n = 43400
n = 11.07
I will assume there is no partial rebate, so the greatest revenue is obtained with 11 rebates
c) looks like another typo, no cost function is shown
simply subtract the cost function from the revenue function obtained above and proceed like I did
To find the function representing the demand for television sets, we need to incorporate the effect of the price and the rebate on the quantity demanded.
a) Let's start by considering the price. We know that the manufacture has been selling 1400 television sets per week at $450 each. This means the price is $450 when 1400 sets are sold. So, we can say that when the price is $450, the demand (number of sets sold) is 1400.
We can assume a linear relationship between price and demand. Let's write the equation of this line in the slope-intercept form: y = mx + b
In our case, demand (y) represents the number of television sets sold per week, and price (x) represents the corresponding price. So, the equation becomes:
Demand = (slope * Price) + Intercept
We can calculate the slope using the given information. When the price decreases by $14 (due to the rebate), the demand increases by 140 sets per week. So, the slope (m) is:
m = (Change in Demand) / (Change in Price)
= 140 sets / $14
= 10 sets / $1
Now, let's find the intercept (b) by substituting the values for demand and price from the given information:
1400 = (10 sets / $1) * $450 + b
1400 = 4500 sets + b
Solving for b, we find:
b = 1400 - 4500
b = -3100
Therefore, the demand function is:
Demand = (10 sets / $1) * Price - 3100
b) To maximize revenue, we need to find the price that will generate the highest total revenue. Total revenue is calculated by multiplying the demand (number of sets sold) by the price.
Let's denote revenue as R and price as P. The demand function we found in part a) can be rewritten as:
Demand = (10 sets / $1) * P - 3100
Substituting this into the revenue equation, we get:
R = (P)*(Demand)
= (P)*[(10 sets / $1) * P - 3100]
Now, we have the revenue function as a quadratic equation in terms of the price. To maximize revenue, we need to find the price that maximizes this quadratic function.
We can do this by taking the derivative of the revenue function with respect to the price (P), setting it equal to zero, and solving for P. The resulting price will maximize the revenue.
I will leave c) for you to answer as it requires considering the cost function and finding the profit-maximizing value of the rebate.