6.A manufacturer of DVD players has been selling 800 each week at 150$each. From a market survey it is discovered that for each $5 reduction in price, they will sell an extra 40DVD players each week.
a)What is the demand function?
b)How large a reduction in price (rebate) should the manufacturer give a buyer so that revenue will be maximised?
the answers are:
a)p(x)=250 - (x/8)
and
b) 25$
a) my profit equation was
P(x) = (800+40x)(150-5x) where x is the number of $5 reductions
b) I differentiated and set that equal to zero to get x=5
so there are 5 reductions of $5 or $25 , as you had
thank you
To find the demand function, we start by identifying the initial price and quantity sold. We know that the manufacturer sells 800 DVD players each week at a price of $150 each.
a) To determine the relationship between price and quantity demanded, we need to find the change in quantity demanded for each change in price. From the market survey, we learn that a $5 reduction in price leads to an increase in sales of 40 DVD players.
Let's calculate the change in price as the new price minus the initial price:
∆P = 150 - (150 - 5) = -5
Now let's calculate the change in quantity as the new quantity sold minus the initial quantity:
∆Q = 800 + 40 - 800 = 40
The slope of the demand function, m, is given by the ratio of the change in quantity to the change in price:
m = ∆Q / ∆P = 40 / -5 = -8
Using point-slope form, where m is the slope and (x1, y1) represents a point on the line, we can write the demand function:
Q - Q1 = m(P - P1)
Substituting the values we have:
Q - 800 = -8(P - 150)
Simplifying the equation gives us:
Q = -8P + 200 + 800
Q = -8P + 1000
But we want to express the demand function in terms of x, the quantity, rather than Q. Since we know that x is equal to Q, we can rewrite the equation:
x = -8P + 1000
To find the demand function, we substitute x for Q and P for price:
p(x) = -8P + 1000
Since we want to solve for P, we can rearrange the equation:
P = (1000 - x) / 8
Thus, the demand function is:
a) p(x) = (1000 - x) / 8
b) To maximize revenue, we need to identify the price that will yield the maximum number of sales. Revenue is calculated by multiplying price (P) by quantity (Q). We can rewrite the equation for revenue as follows:
R = P * Q
Substituting the demand function p(x) into the revenue equation gives us:
R = (P * x) = (P * (1000 - x)) / 8
To find the value of x that maximizes revenue, we take the derivative of R with respect to x and set it equal to zero:
dR/dx = (1000 - 2x) / 8 = 0
Simplifying the equation gives us:
1000 - 2x = 0
Solving for x gives us:
x = 500
Now, substitute the value of x into the demand function to find the corresponding price:
P = (1000 - x) / 8
P = (1000 - 500) / 8
P = 125
Therefore, to maximize revenue, the manufacturer should offer a $25 reduction in price (rebate) to the buyer, resulting in a final price of $125 for each DVD player.
b) The manufacturer should provide a $25 reduction in price (rebate) to maximize revenue.