I need real help on how to find the vertex for this question! PLS EXPLAIN

Question

The starting price of grapes per mz is $8.00, at which you can sell 60 m^2 worth- For every $3.00 increase in price, you sell 3 m^2 less. So use the same method you just used to find the price of grapes per m' of area in order to maximize revenue.

number of price increases x=

price of grapes per m^2=

Answer 1

Hi Ariel G.,

So I presume your difficulty is in trying to find the total sales revenue.

So let's start by assigning P = price per m^2 (probably should be per m^3, who would buy grapes by area?)

and the volume sold as V. Then revenue R = sales = P*V. R will be your dependent variable (y) and P will be your independent variable (x), if you care to think in graphical terms.

Now V = 60 - 3*(P-8)/3 . Why? 60 represents the "starting" volume (60 m^3). The volume sold thereafter falls off by -3 (the volume sold less), and (P-8)/3 is the scaling factor for the price increases per m^3 ABOVE the baseline case of $8, in increments of $3. You should plug in some values mentally, just to see how that works, so that you can do it next time!

You want revenue R.

So that's R = P*V = P*(60-(P-8)) or P*(68-P) . That's your quadratic, and you know already how to go about finding the vertex point from your quadratic coefficients, don't you? (Note that there isn't a constant c in the quadratic, so c=0. That simplifies finding your 2 roots, and you can plug in the P value midway between those to solve for the vertex revenue, if you've forgotten the vertex formula).

By the way, you may want to note for future reference, that sometimes prescription drugs are dosed by patient body volume or mass (/kg or /m^3) and sometimes by body surface area (/m^2). So get in the habit of always checking your units in problems! An 80 kg patient and a 2 m^2 patient are QUITE different calculations!

-- Cheers, -- Mr. d.