# Thread: fucntions and graphs problem solving

1. ## fucntions and graphs problem solving

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 2. 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 ($\displaystyle n$) and the selling price ($\displaystyle x$) is linear, because for every $4 decrease in the selling price, the number sold increases by the same amount,$\displaystyle 20$. So, for some constants$\displaystyle A$and$\displaystyle B$:$\displaystyle n = A + Bx$When$\displaystyle x = 100, n = 50$; and when$\displaystyle x = 96, n = 70$. Therefore:$\displaystyle 50= A + 100B\displaystyle 70 = A + 96B$Solve these equations to find the values of A and B. Thus, the relationship between$\displaystyle n$and$\displaystyle x$is:$\displaystyle n = 550-5x$(b) The revenue from the sales is the number of chairs sold multiplied by the selling price per chair. So:$\displaystyle R = nx$Now use the answer to (a) and you're there. (c) Each chair costs$40 to buy. So the total cost of $\displaystyle n$ chairs is $...? Now use part (a) to write this in terms of$\displaystyle x$. Then subtract this total cost from the total revenue in part (b) to find the total profit$\displaystyle P(x)$in terms of$\displaystyle x$. Can you complete it now? Grandad 3. im sorry, but i still dont fully understand part c. i was wondering whether you could please show the working out? thank you very much 4. Hello Tessarina Originally Posted by Tessarina im sorry, but i still dont fully understand part c. i was wondering whether you could please show the working out? thank you very much The cost of$\displaystyle n$chairs at$\displaystyle 40$dollars per chair is$\displaystyle 40n\displaystyle =40(550-5x)$, from part (a)$\displaystyle =(22000 - 200x)$Now the profit is the total revenue minus the total cost. So, given that the revenue is$\displaystyle (550x - 5x^2)$the profit is$\displaystyle P(x) = (550x -5x^2) - (22000-200x)\displaystyle = 550x -5x^2 - 22000+200x)\displaystyle =750x-5x^2-22000$Grandad 5. I'm really sorry to ask again, but i was wondering whether you could also show me the whole working out for part b? thank you 6. Hello Tessarina Originally Posted by Tessarina I'm really sorry to ask again, but i was wondering whether you could also show me the whole working out for part b? thank you The total revenue is the number of items sold multiplied by the selling price per item. If we translate this into symbols, this is:$\displaystyle R = nx$...(1) because$\displaystyle R =$total revenue$\displaystyle n =$number of chairs sold$\displaystyle x =$selling price of one chair But we showed in part (a) that:$\displaystyle n = 550 - 5x$So in equation (1), we can replace$\displaystyle n$by$\displaystyle (550 - 5x)$to get:$\displaystyle R = (550-5x)x$and, if we remove the brackets:$\displaystyle R = 550x-5x^2\$
OK now?