[2 marks] Fashion One Chain Store found that t weeks after the end of a sales promotion the volume of sales was given by

Question

[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.

Answer 1

To find the decay constant k and the sales volume at the end of the fourth week, we can use the given information and equations. Let's break down the steps:

a) Finding the decay constant k:
We have the equation S(t) = B + Ae^(-kt), where S(t) represents the sales volume at time t.

Given that B = 50,000, S(1) = $83,515, and S(3) = $65,055, we can substitute these values into the equation:

For t = 1, S(1) = 50,000 + Ae^(-k * 1)
83,515 = 50,000 + Ae^(-k)

Similarly, for t = 3, S(3) = 50,000 + Ae^(-k * 3)
65,055 = 50,000 + Ae^(-3k)

We now have a system of two equations with two unknowns (A and k). We can solve it to find the value of k.

Subtracting the first equation from the second equation:
65,055 - 83,515 = 50,000 + Ae^(-3k) - (50,000 + Ae^(-k))
-18,460 = Ae^(-3k) - Ae^(-k)

Since the sales volumes are decreasing exponentially, Ae^(-3k) < Ae^(-k). Therefore, we can rewrite the equation as:

-18,460 = -A(e^(-k) - e^(-3k))

Dividing both sides by -A and simplifying:
18,460/A = e^(-k) - e^(-3k)

Now, let's introduce a variable: x = e^(-k)

Using the property that e^(-3k) = (e^(-k))^3, we can rewrite the equation as:
18,460/A = x - x^3

Simplifying further:
x^3 - x + (18,460/A) = 0

Now we have a cubic equation. We need to solve this equation to find the value of x, and then substitute it back into x = e^(-k) to find k.

b) Finding the sales volume at the end of the fourth week:
Once we have determined the decay constant k, we can use it to calculate the sales volume at the end of the fourth week.
For t = 4, we substitute the values into the original equation:
S(4) = 50,000 + Ae^(-4k)

Now that we have outlined the steps, you can follow them to find the answer to both parts of the question.