[2 marks] Fashion One Chain Store found that t weeks after the end of a sales promotion the volume of sales was given by
S(t)BAekt (0t4)
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) Fin the decay constant k.
b) Findthesalesvolumeattheendofthefourthweek.
a) To find the decay constant k, we can use the given information about the sales volumes at the end of the first and third weeks.
We know that S(1) at the end of the first week is $83,515, so we can substitute the values into the formula:
83,515 = 50,000 + Ae^(-k*1)
Similarly, for the sales volume at the end of the third week:
65,055 = 50,000 + Ae^(-k*3)
We now have a system of two equations with two unknowns (A and k). We can solve this system to find the value of k.
Subtracting 50,000 from both sides of both equations, we get:
33,515 = Ae^(-k*1)
and
15,055 = Ae^(-k*3)
Dividing the two equations, we have:
(33,515 / 15,055) = (Ae^(-k*1) / Ae^(-k*3))
Simplify the equation:
2.225 = e^2k
Take the natural logarithm (ln) of both sides:
ln(2.225) = 2k
Divide both sides by 2:
k = ln(2.225) / 2
Calculate this expression using a calculator or computer:
k ≈ 0.395
b) To find the sales volume at the end of the fourth week (t = 4), we can substitute the values into the formula:
S(4) = 50,000 + Ae^(-0.395 * 4)
We still need to find the value of A. To do this, we can substitute the sales volume at the end of the first week:
83,515 = 50,000 + Ae^(-0.395 * 1)
Solve this equation for A:
33,515 = Ae^(-0.395)
Divide both sides by e^(-0.395):
A = 33,515 / e^(-0.395)
Calculate this expression using a calculator or computer, then substitute the values into the equation for S(4):
S(4) = 50,000 + Ae^(-0.395 * 4)
Evaluate this expression to find the sales volume at the end of the fourth week.