Question

The weekly sales in thousands of items of a product has a seasonal sales record approximated by n=84.59+24.8 sin pi t/24 (t is time in weeks with t = 1 referring to the first week in the year). During which week(s) will the sales equal 96,990 items?
A. week 4, week 20, and week 52
B. week 21 and week 30
C. week 4 and week 47
D. week 30 and week 47

Answer 1

@agent0smith

Answer 2

@Michele_Laino ? help on this one pls ?

Answer 3

Please you have to solve the subsequent equation:
\[84.59+24.8*\sin(\frac{ \pi t }{ 24 })=96.99\]
adding to both sides of the above equation the quantity -84.59, and dividing both sides the resultant equation, we get:
\[\sin(\frac{ \pi t }{ 24 })=\frac{ 1 }{ 2 }\]
now, generally we have that sin x=0.5 when x=pi/6, so we get:
\[\frac{ \pi t }{ 24 }=\frac{ \pi }{ 6 }\]
from which t=4 weeks.

Answer 4

Now, the solution above is the first solution and furthermore it is not the mor general solution, the general solution is:
\[\frac{ \pi t }{ 24 }=\frac{ \pi }{ 6 }+2k \pi\]
because sin x is a periodic function and its period is equal to 2*pi, furthermore k is an integer with sign. So from the last equation we get as solution for t the subsequent expression:
\[t=\frac{ 24 }{ \pi }*(\frac{ \pi }{ 6 }+2k \pi)=4+48 k\].
Now letting k=1, we have:
\[t=4+48=52\]
As I said befor there are two solution, that it's evident from the subsequent graph:

Answer 5

So the second solution is:
\[\frac{ \pi t }{ 24 }=\frac{ 5 \pi }{ 6 }+2k \pi\]
like before I go to solve that equation for t:
\[t=\frac{ 24 }{ \pi }*(\frac{ 5 \pi }{ 6 }+2 k \pi)=20+48 k\]
setting k=0, we get:
\[t=20.\]
Reassuming the solutions that we got are: 4, 20 and 52, so your answer is A.

Answer 6

thank you !!!!: )

Answer 7

Thank you!