At the end of the holiday season in January, the sales at a department store are expected to fall. It is estimated that for the x day of January the sales will be S (x) = 5 + 25 / (x + 1) ^2 ( a) Find the total sales for January 11 and determine the rate at which sales are falling on that day. ( b) Compare the rate of change of sales on January 4 to the rate on January 11. What can you infer about the rate of change of sales?

Question

Mathematics

Kendall Randolph

15 October, 12:55

At the end of the holiday season in January, the sales at a department store are expected to fall. It is estimated that for the x day of January the sales will be S (x) = 5 + 25 / (x + 1) ^2 ( a) Find the total sales for January 11 and determine the rate at which sales are falling on that day. ( b) Compare the rate of change of sales on January 4 to the rate on January 11. What can you infer about the rate of change of sales?

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Ivan Richard · Answer 1

See below

Step-by-step explanation:

a) The total sales for January 11 are the value S (11). This is:

S (11) = 5+25 / (12) ²=745/144=5.173611111 ...

The rate at which sales fall in Jan 11 is the derivarive S' (x) in x=11. We have that S' (x) = -50 / (x+1) ³ (by usual laws of derivatives), hence the rate is S' (11) = -50 / (12) ³=-0.0289351851 ...

A negative derivative implies that the original function S is decreasing. This is consistent with the information given; the sales S (x) are falling.

b) Let's compare S' (11) = -0.02893518518 ... and S' (4) = -50 / (5) ³=-0.4. We have that S' (4) = -0.4<-0.02893518518 ... = S' (11). Therefore the rate of change is increasing towards 0, which means that sales will decrease to 5 until they stabilize (rate of change equal to zero).