maths help please

royalties

Clothing company produces one type of shirt. Research has found that if company prices shirts at 30 € each they will sell 500 units per week. Also if price was 55€ they sell none. The company prices shirts at x euro each where x is greater than or equal to 30 and less than or equal to 55.
Q: Find an expression for sales per week in terms of x?
q: write an eqn for the value in euro of weekly sales in terms of x?

ForestFire

I am know Maths expert, but.....

From the details of the question I don't think it is really possible unless...
Are we to assume that the sales drop off is linear from 30€ to 55€?
This is highly unlikly but the only way to get an equation?


Therefore

(30,500)
(35, 400)
(40,300)
(45, 200)
(50, 100)
(55,0)

Now any two points on a graph that is linear it is possible to get equation, but I can't remeber this...sorry

Yakuza

If there's a (negative) linear relationship between price and sales, what would the slope of the line be?

We are assuming a formula of the form y = mx +c, where y is the dependent variable (number of sales), m the slope of the line, x the independent variable (price) and c the y-intercept.

What is the slope of this line? The slope of the line is the change in y over the change in x. In going from a price of €30 to €55, sales decrease from 500 to zero. Therefore y increases by -500 for a change of +25 in x, so the slope is -20.

The line is therefore in the form of y = -20x + c

To find c, plug in either of the (x,y) pairs you have (30,500) or (55,0) and solve for c. I'm not going to give the entire answer here, it's against the spirit and charter of the forum. Post up your answer and we can give more feedback.

For your second question, revenue = price by sales. (or x * y).
Once you have an equation for y in terms of x, multiply it by x again and this will give you an equation for the weekly revenue.

royalties

so sales can be defined as
y (sales)= -20x+1100 and represents a linear function

and revenue as:
sales by price= x(-20x+1100)= -20x^2+1100x.

is there a method for maximising sales and revenue or something

Yakuza

Yep, Sales(y) = 1100-20x and revenue (x*y) is 1100x - 20x²

To maximise your revenue, find the first deriviative of the revenue function, set it equal to zero and solve for x. (To be sure it is a maximum, look at the second derivative, if it's negative (which it will be), then we're talking about a local maximum).

In this case, the maximum of the function falls outside (below) the price range (assuming the minimum price is €30), so maximum revenue will be found at a price of €30. Generally, it is not a good idea to extrapolate beyond known ranges, hence the answer isn't strictly the maximum of the function.