Question

A sportswear store sells baseball caps with the local baseball team's logo on them. Last year, the store sold 600 caps at $15 each. The store manager is planning to increase the price. A consumer survey shows that for every $1 increase, they will be a drop of 30 sales a year.

Answer 1

a) What should the selling price be to maximize the annual revenue?
b) What is the maximum annual revenue from the caps?

Answer 2

The answer for a) is supposed to be: 84.5 and b) 6.5, 6.5. How?

Answer 3

if the answer for A is 84.5, it is against on the survey. .

Answer 4

given that the price per cap is 15, and survey states that their will be a drop of sales by 30 number of caps if they will increase 1 per cap, how could the answer be 84.5?

i think the answer for the increase is less than 1 dollar. that will be 15+x; where x is less than 1

that's my interpretation to the problem. .

Answer 5

Let R be revenue per year, in dollars.
R = units sold multiply price per unit.
Let units sold be U caps.
Let price per unit be P, in dollars per cap.
Therefore:
R = UP.

to begin with, P = $15 per cap, and U = 600 caps.

To increase the price by $1, and drop sales by 30:
R = (600 - 30) (15 + 1).

To do the same thing again:
R = (600 - 30 - 30) (15 + 1 + 1) = (600 - 60)(15 + 2).

Notice that these will decrease by multiples of 30 and increase by multiples of 1 respectively.

Give the number of repetitions the value n.
So:
U = 600 - 30n.
P = 15 + n.

So:
R = UP = (600 - 30n) (15 + n).
Expanding:
R = 9000 + 600n - 450n - 30n^2.
R = 9000 + 150n - 30n^2.
Factorising:
R = -30 [n^2 - 5n - 30 ].

Completing the square:
R = -30 [ (n - 2½)^2 - (2½^2) - 30 ].
R = -30 [ (n - 2½)^2 - 36¼ ].
R = (30)36¼ - 30(n - 2½)^2.

Since you have completed the square, any derivative will only be 0 at the point when the square is 0.
i.e. (n - 2½)^2 must equal 0.
(n - 2½)^2 = 0.
Square rooting:
n - 2½ = 0.
n = 2½.

Differentiating:
dR/dn = -60(n - 2½) = 150 - 60n.
Solving for 0:
dR/dn = 0 → 150 - 60n = 0.
150 = 60n.
n = 15/6 = 2½.

So, for (a) :
Let n = 2½.
P = 15 + n = 15 + 2½ = 17½.
Thus P = $17.50 per cap.

For (b) :
Again, n = 2½.
R = (600 - 30n) (17½).
R = (600 - 30(2½)) (17½).
R = (600 - 75) (17½).
R = (525)17½.
R = $9187.50 per year.

Answer 6

woah. That's how to get the answer?