Question

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?

Answer 1

See below

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).