im not sure where to go from here. can someone help me? IM IN DESPERATE NEED

Question

Central Perk Coffee Shop sells a specialty drink at $2.60 per cup. The shop sells about 200 cups per day. Customer surveys show that for every $0.05 decrease in price, the shop will sell 10 more cups per day. At what coffee price(s) will they earn a profit of $630?

Okay so I found the vertex which is (16, 648) but I'm not sure where I should input 630 to receive the coffee prices.

Answer 1

I m not sure how much help you need, given that you have the vertex.

You just let f(P) = 630 and solve for P

Assuming you had f(P) or, apparently, f(N) where N is the number of nickels that price is reduced, this should be straightforward. Since 630 is not the peak, you should have two solutions.

I solve here using P, but transfer to N change to, possibly, match your solution.

This problem does not define profit per cup, but price.

Thus, I have to assume that $2.60 is the profit per cup.

Revenue = PQ

We have (P,Q) = (2.6,200)

If, for every $.05 decrease, quantity increases by 10, m = 10/-.05 = -200

Q = mP + q-intercept

200 = -200(2.6) + q-intercept

200 = -520 + q-intercept

720 = q-intercept

Q = -200P + 720

R = PQ = (-200P + 720)P = -200P^2 + 720P

Completing the square to find the vertex,

-200P^2 + 720P = -200(P^2 - 3.6P) = -200(P^2 - 3.6P + 3.24) + 200 * 3.24 =

-200(P - 1.8)²+ 648

The vertex is (1.8, 648) Your 16 must refer to the number of nickels that the price would be reduced - (2.6 - 1.8)/.05 = 16, so your function may look a little different. If I make P = 2.6 - .05N, the equation above becomes

-200(2.6 - .05N - 1.8)² + 648 = -200(.8 - .05N)² + 648 = -1/2(16 - N)² + 648

If I use your equation, 630 is the revenue, so I can either solve using the vertex form or the standard form.

Using the vertex form, -1/2(16 - N)² + 648 = 630

-1/2(16 - N)² = -18

(16 - N)² = 36

16 - N = ±6

N = 10, 22

P = 2.6 - 10(.05), 2.6 - 22(.05)

P = 2.10, 1.50

If you use the standard form rather than vertex form to solve, you solve using factoring or the quadratic formula.

If you had used my version, instead, you would have used

-200(P - 1.8)²+ 648 = 630

-200(P - 1.8)² = -18

(P - 1.8)² = .09

P - 1.8 = ± .3

P = 1.8 - .3, 1.8 + .3

P = 1.50, 2.10

Again, you could have gone back to the standard form, -200P^2 + 720P, and solved

-200P^2 + 720P = 630

-200P^2 + 720P - 630 = 0

20P^2 - 72P - 63 = 0

(2P - 3)(10P - 21) = 0

P = 1.50, P = 2.10

If you have done all the work to get it in vertex form, though, you have basically done the quadratic formula, so you might as well use this, but do whichever you prefer.

Answer 2

(2.6-.05)*(200+10)=535.5

What is the starting profit/container?