Metro Department Store found that t weeks after the end of a sales promotion the
ID: 3191674 • Letter: M
Question
Metro Department Store found that t weeks after the end of a sales promotion the volume of sales was given by S(t) = B +Ae^(-kt) where t is greater than or equal to 0 or less than or equal to 4 where B=50,000 and is equal to the average weekly volume of sales before the promotion. The sales volumes at the end of the first and third weeks were $83,515 and $65,055, respectively. Assume that the sales volume is decreasing exponentially. a.) Find the decay constant k. b.) Find the sales volume at the end of the fourth week. c.) How fast is the sales volume dropping at the end of the fourth week?Explanation / Answer
B = 45000 (a constant amount, so you can plug it in) S(1) = 83580 and S(3) =66870 are two data points. Let's see what happens when we plug that in: (a) 83580 = 45000 + Ae^(1k) 38580 = Ae^k.............Eq1 66870 = 45000 + Ae^(3k) 22870 = Ae^(3k).............Eq2 Divide Eq1 by Eq2: 1.764 = e^(-2k) ln 1.764 = -2k k = (ln 1.764)/-2 = -.2838 We need to find A also, so I will use Eq1 for this: 38580 = Ae^(-.2838) 38580/e^(-.2838) = A = $51240.79 Thus, the function is S(T) = 45000 + 51240.79e^(-.2838T) (b) S(4) = 45000 + 51240.79e^(-.2838*4) = $61,466.63 (c) S' (T) = -.2838*51240.79*e^(-.2838T) S '(T) = -14542.14e^(-.2838T) S '(4) = -14542.14e^(-.2838*4) = -$4,673.23 / week