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 (0 ≤ t ≤ 4) where b = 48,000 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 $80,360 and $63,200, respectively. Assume that the sales volume is decreasing exponentially. (a) Find the decay constant k. (Round your answer to five decimal places.) a) k = _______ (Fill in the blank) (b) Find the sales volume at the end of the fourth week. (Round your answer to the nearest whole number.) b) $_______ (Fill in the blank)

Answer 1

The decay constant k is approximately 0.12329 while the sales volume at the end of the fourth week is about $57,811.

Step-by-step explanation:

We are given that the equation governing the volume of sales is s(t) = b ae-kt where b = 48,000, and that the sales volumes at the end of the first and third weeks were $80,360 and $63,200, respectively. Firstly, let's find a

For the first week, we obtain equation 80,360 = 48,000a e-k.

This simplifies to a e-k = 1.67458333(1)

For the third week, we obtain the equation 63,200 = 48,000a e-3k.

This simplifies to a e-3k = 1.31666667(2)

Dividing (2) by (1) gives us e-2k = 0.786764706,

so k = -ln(0.786764706) / 2 = 0.12328767.

We substitute k back into (1) to get a

= 1.67458333 / e-0.12328767

= 1.879563.

For the fourth week,

s(4) = 48,000(1.879563) e-0.12328767*4

= 57,811 (rounded to the nearest dollar).

Learn more about Decay Constant and Volume Sales