fucntions and graphs problem solving

**Tessarina** · April 25th 2010, 08:20 PM

A retail furniture shop can buy dining room chairs from a manufacturer for $40 each. The retailer knows that if each chair sells for $100, 50 of them can be sold per week. However the research department believes that for every $4 reduction in price they will sell 20 more chairs per week.
Suppose that n chairs are sold per week, that each chair sells for $x and that the total sales revenue is $R per week.

a) show that n=550 -5x

b) show that R= 550x -5x^2

c) show that the profit function, P(x) = 750x -5x^2 -22000

**Grandad** · April 26th 2010, 01:26 AM

Hello Tessarina

Originally Posted by Tessarina

A retail furniture shop can buy dining room chairs from a manufacturer for $40 each. The retailer knows that if each chair sells for $100, 50 of them can be sold per week. However the research department believes that for every $4 reduction in price they will sell 20 more chairs per week.
Suppose that n chairs are sold per week, that each chair sells for $x and that the total sales revenue is $R per week.

a) show that n=550 -5x

b) show that R= 550x -5x^2

c) show that the profit function, P(x) = 750x -5x^2 -22000

(a) The relationship between the number sold ( $n$ ) and the selling price ( $x$ ) is linear, because for every $4 decrease in the selling price, the number sold increases by the same amount, $20$ . So, for some constants $A$ and $B$ :

$n = A + Bx$

When $x = 100, n = 50$ ; and when $x = 96, n = 70$ . Therefore:

$50= A + 100B$

$70 = A + 96B$

Solve these equations to find the values of A and B. Thus, the relationship between $n$ and $x$ is:

$n = 550-5x$

(b) The revenue from the sales is the number of chairs sold multiplied by the selling price per chair. So:

$R = nx$

Now use the answer to (a) and you're there.

(c) Each chair costs $40 to buy. So the total cost of $n$ chairs is $...?

Now use part (a) to write this in terms of $x$ .

Then subtract this total cost from the total revenue in part (b) to find the total profit $P(x)$ in terms of $x$ .

Can you complete it now?

Grandad