The initial price of buzzcom stock is 15 per share After 20

The initial price of buzz.com stock is $15 per share. After 20 days the stock price is $30 per share and after 40 days the price is $40 per share. Assume that while the price of the stock is not zero it can be modeled by a quadratic function.

Find the multipart function s(t) giving the stock price after t days. s(t) = {-1/160 t^2 + 7/8 t + 15 -1/160 t^2 + 7/8 t + 15 0 lessthanorequalto t lessthanorequalto 155 t > 155 If you buy 1000 shares after 30 days, what is the cost? (b) To maximize profit, when should you sell shares? How much will the profit be on your 1000 shares purchased in (a)?

Solution

(a)Let the quadratic function representing the price of buzz.com stock be s(t) = at²+bt +c, where a,b,c are arbitrary real numbers and t is the number of days after purchase. Since the initial price of is $15 per share, hence on substituting t = 0 , we get c = 15. Then, s(t) = at²+bt +15. Further, when t = 20, s(t) = 30 so that 30 = a(20)²+ 20b+15 or, 400a +20b = 30-15 = 15 or( on dividing both the sides by 5) ,80a+4b = 3…(1). Also, when t = 40, s(t) = 40 so that 40 = a(40)²+40b+15 or, 1600a+40b = 40-15 = 25, or(on dividing both the sides by 5) , 320a +8b = 5…(2). On multiplying the 1^st equation by 2 and then subtracting the result from the 2^nd equation, we get 320a +8b -160a -8b = 5-6 or, 160 a = -1 so that a = -1/160. Then, from the 1^st equation, we have 80(-1/160) +4b = 3 or, -1/2 +4b = 3 or, 4b = 3+1/2 = 7/2 so that b = 7/8. Then the quadratic function representing the price of buzz.com stock is s(t) = (-1/160)t²+(7/8)t +15.

The cost of 1000 shares bought after 30 days is 1000*[- (30)²/160 +7(30)/8 +15] = 1000*[ -900/160+ 210/8+15] = 1000[ -45/8 + 210/8 +15} = 1000* 285/8 = $ 35625.

(b ). For maximizing profit, ds/dt = 0 and d²s/dt² should be negative. Here, ds/dt = -2t/160 +7/8 so that ds/dt = 0 when 2t/160 = 7/8 or, t = (7/8)*160/2 = 70 days. Also, d²s/dt² = -2/160 = -1/80 is negative regardless of the value of t. Hence, the profit is maximum if the shares are sold after 70 days.

The profit on 1000 shares sold after 70 days will be 1000*[- (70)²/160 +7(70)/8 +15]- 1000*15 = 1000[ -4900/160+ 490/8 + 15-15] = 1000( -4900/160 +490/8) = 1000[( -4900 +9800)/160] = 1000*4900/160 = $ 30625.