15. 1/3 points | Previous Answers SCalcET8 4.7.502.XP. My Notes A manufacturer h

Question

15. 1/3 points | Previous Answers SCalcET8 4.7.502.XP. My Notes A manufacturer has been selling 1000 flat-screen TVs a week at $300 each. A market survey indicates that for each $10 rebate offered to the buyer, the number of TVs sold will increase by 100 per week. (a) Find the demand function (price p as a function of units sold x) p(X) = 10 (b) How large a rebate should the company offer the buyer in order to maximize its revenue? $350 (c) If its weekly cost function is C(x) = 65,000 + 150x, how should the manufacturer set the size of the rebate in order to maximize its profit? $175 Need Help? Read itWatch I Talk to a Tutor Submit Answer Save Progress Practice Another Version

Explanation / Answer

(a) let
P is the price per TV.
X is the number sold per week.
R is the number of $10 rebates offered.
First, the price is $300. That price decreases by $10 for each rebate offered.

P = 300 - 10R
Secondly, the number sold is 1000. Unless of course, some rebates are offered. Then the number sold goes up by 100 for each Rebate!

X = 1000 + 100R

X - 1000 = 100R
X/100 - 10 = R

Plugging it in:
P = 300 - 10(X/100 - 10)
P = 300 - (X/10 - 100)
P = 300 - X/10 + 100
P = 400 - X/10 This is price (P) as a function of units sold (x).

(b) Maximize the revenue
revenue = PX

Now we should substitute two equations in for P and X. We should use equations that have R (number of rebates) in them
revenue = [300 - 10(R)][1000 + 100(R)]
revenue = 300,000 + 30,000(R) - 10,000(R) - 1000(R^2)
revenue = 300,000 + 20,000(R) -1000(R^2)

It asks for the maximum revenue. Let's take the derivative of both sides, and solve for zero. The last function graphs out a parabola with its curve on top. We're looking for that top point of the curve. Well, that point will be where the slope is zero. When you take the derivative, you come up with an equation for slope. When you make slope become zero, you have revenue max.

0 = 20,000 - 2000(R)

2000(R) = 20,000
R= 10

Remember, R is the number of ten dollar rebates. So
Optimal rebate amount = $100

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.