optimizing sale problems

Question

if a bakery sells sandwiches for $8, they sell an average of 360 per day.  they have determined that if they increase the the price by $1, they will sell 20 less sandwiches per day.  what is the optimal price of the sandwich to produce the greatest amount of money

Answer 1

Let total amount earned per day be T, and x be price increase.

T
= (8 + x)(360 - 20x)
= 2880 + 360x - 160x - 20x^2
= -20x^2 + 200x + 2880

T' = -40x + 200

Setting T' to zero, we have:

-40x + 200 = 0
40x = 200
x = 200 / 40
x = 5

T'' = -40 < 0, so we have a local max at x = 5.
Since T is a quadratic equation, this means that this local max is also an absolute max.

8 + x = 8 + 5 = 15

Hence, optimal proce is $13