[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)BAekt (0t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) Fin the decay constant k.
b) Findthesalesvolumeattheendofthefourthweek.

To find the decay constant (k) in the given equation, we can use the information given about the sales volumes at the end of the first and third weeks.

a) Find the decay constant (k):
We are given two data points:
S(1) = $83,515 (sales volume at the end of the first week)
S(3) = $65,055 (sales volume at the end of the third week)

Substituting these values into the equation:
S(1) = B + Ae^(-k * 1) = 50,000 + Ae^(-k)
S(3) = B + Ae^(-k * 3) = 50,000 + Ae^(-3k)

Now, we have two equations with two unknowns (A and k). Subtracting the first equation from the second equation, we can eliminate A:
S(3) - S(1) = 50,000 + Ae^(-3k) - (50,000 + Ae^(-k))
Simplifying:
$65,055 - $83,515 = Ae^(-3k) - Ae^(-k)
-$18,460 = Ae^(-k)(e^(-2k) - 1)

Since e^(-k) is never zero, we can divide both sides of the equation by Ae^(-k):
-($18,460 / Ae^(-k)) = e^(-2k) - 1

Let's denote the ratio -($18,460 / Ae^(-k)) as R:
R = e^(-2k) - 1

Now, we can solve for R:
R = e^(-2k) - 1
R + 1 = e^(-2k)
Taking the natural logarithm of both sides:
ln(R + 1) = -2k
Simplifying:
k = -ln(R + 1) / 2

Therefore, the decay constant (k) can be found using the formula k = -ln(R + 1) / 2, where R = -($18,460 / Ae^(-k)).

b) Find the sales volume at the end of the fourth week:
Using the equation S(t) = B + Ae^(-kt), where t = 4 (end of the fourth week), and substituting the given values:
S(4) = 50,000 + Ae^(-k * 4)

Now that we have found the decay constant (k) in part a), substitute the value of k into the equation:
S(4) = 50,000 + Ae^(-k * 4)

To find the value of A, we need another data point. Unfortunately, we don't have this information, so we cannot determine the exact sales volume at the end of the fourth week without another piece of data.