Jessica sells T-shirts from her small kiosk in the shopping mall. On average, sh

Question

Jessica sells T-shirts from her small kiosk in the shopping mall. On average, she sells 30 T-shirts per day at a price of $19.00 each. She estimates she will sell 3 additional shirts each day for every $1 reduction in the price per T-shirt. Find each of the following: 29. A modeling Jessica's daily revenue as related to reduction in price. 30. The range of prices which would allow her to maintain or increase her current revenue. 31. The maximum amount by which she can increase her daily revenue.

Explanation / Answer

Let the price be reduced by $ x. Then the number of T-shirts sold daily is 30+3x and the price of a T-shirt is $(19-x).

29. The daily revenue of Jessica is R(x) = (30+3x)(19-x) = 570+27x- 3x2.

30. R(x) =- 3x2+27x+570 =-3(x2-9x)+570 =-3(x2-2*9x/2+81/4)+570+3*81/4=-3(x-9/2)2 +2523/4. This is the equation of a parabola opening downwards with vertex at ( 9/2,2523/4). Since the vertex is the highest point of such a parabola, therefore, Jessica’s revenue will keep increasing if x 9/2. Thus, the range of prices which will maintain or keep increasing Jessica’s daily revenue is [19-x,19] with x 9/2, i.e. [14.50,19].

31. The maximum amount by which Jessica can increase her daily revenue is (30+3*9/2)(19-9/2) -30*19 = 43.5* 14.5 - 570 = 630.75- 570 = $ 60.75.

Note:

There seems to be an error here. The number of T-shirts has to be whole number while 30+3*9/2 = 43.5 is NOT a whole number. It may be rounded off to either 43 or 44 and the answer may be amended accordingly.

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