# Math Help - Linear Algebra - I'm Stuck ona Question

1. ## Linear Algebra - I'm Stuck ona Question

Hey Guys, I'm kind of embarrassed to ask this question but I've attempted it a few times now and can't afford to waste anymore time on it. I just can't seem to start it properly I think I've over thought it. anyways here is the question verbatim:

Based on her market research, Jill has discovered that the number of glasses of lemonade sold in a day is a function of the sale price (p=p(x), where x is the price, in cents, of a glass of lemonade).

a.) Jill noticed that 75 glasses of lemonade were sold when the price of a glass was set to 80 cents. If Jill assumes that the number of glasses sold is a linear function of the sale price, what should p(x) be?

Well good luck guys. I hope someone can push me in the right direction

regards,

2. well... the number of sold glasses = y = 75.
the price of a glass of lemonade = x = 80.
linear function: y = k*x + m.
so all we know is 75 = k*80 + m.
that equation has an infinte number of equations... y = x - 5, y = 2x - 85, y = 3x - 165... i guess the right answer is the functions y = n*(x - 80) + 75, where n is an integer...

edit: don't be embarassed, it's a lousy question since there are an infinite number of possible answers...

3. Hmm, that clears up a few things. I should have put the second part of question on here. It changes and elaborates on a. I guess the question was trying to make someone create a generic equation like y = n*(x - 80) + 75, then use that in the second part b.

b) Jill noticed that 75 glasses of lemonade were sold when the price of a glass was set to 60 cents, but only 45 glasses were sold if the price was set to 80 cent (hint: value per glass has changed.) If Jill assumes that the number of glasses sold is a linear function of the sale price, what should p(x) be?

could we answer part b this way:

Eq1 y = n*(x-60) + 75
Eq2 y = n*(x-80) + 45?

4. well, no because now you have to make them into one equation, and given two points there's always one unique solution.
k = $\frac{\Delta y}{\Delta x}$
p1 = (60, 75), p2 = (80, 45)
$\Delta y$ = 45 - 75 = -30
$\Delta x$ = 80 - 60 = 20
k = -30/20 = -1.5
now we know that 75 = -1.5*60 + m... m = 75 + 90 = 165.
so the function in b is y = -1,5x + 165

5. ## Ugghh

Hey r0r0trog, I hope your around to answer more questions! I thought I'd be able to figure out the rest of this question once I got it started thanks to you. However I was mistaken.

Carrying on after B, there is more:

The gross profit f(x) of her lemonade stand is equal to the number of glasses sold, multiplied by the sale price. It costs 20cents to get the ingredients for one glass of lemonade and mix them together. It also costs 2 dollars a day to keep the lemonade stand up.

c) Sketch the gross profit f(x) and the overhead (as a function of price) on the same graph. Find an expression for the net profit expressed as a function of the sale price

d) At what price should Jill sell her lemonade to maximize her profit? At what price should Jill sell her lemonade to make any profit at all?

Ok this is what I came up with for C
net price = (p(x)*20)+200, but its not giving me any realistic answers

This question has become my nemesis question! Thanks for anymore help