Question

2.

The monthly sales S (in hundreds of units) of baseball equipment for an Internet sporting goods site

are approximated by

77

S=56.9–40.7cos

6

where t is the time in months), with t=1 corresponding to January. Determine the months when

sales exceed 7700 units at any time during the month.

O May through September

O March through August

O March through September

O April through August

O August through April

Answer 1

March through August

Explanation:

Ok, in order to solve this problem, we must start by building an equation to solve. The original equation was:

`S=56.9-40.7cos ((\pi)/(6)t)`

and we need to figure out the months when the sales exceed 7700 units. Since the equation is given in hundreds of units, we need to divide those 7700 units into one hundred to get 77 hundred units. So we can go ahead and substitute that value in the equation:

`77=56.9-40.7cos ((\pi)/(6)t)`

if you wish you can rewrite the equation so the variable is on the left side of it but it's up to you. So you get:

`56.9-40.7cos ((\pi)/(6)t)=77`

and now we solve for t

`-40.7cos ((\pi)/(6)t)=77-56.9`

`-40.7cos ((\pi)/(6)t)=20.1`

`cos ((\pi)/(6)t)=(20.1)/(-40.7)`

`cos ((\pi)/(6)t)=-0.4938`

`(\pi)/(6)t=cos^(-1)(-0.4938)`

`(\pi)/(6)t=2.087`

`t=(2.087(6))/(\pi)`

`t=3.98 months`

but there is a second answer to this problem. Notice that the function cos can be 2.87 at
`2\pi-2.087=4.1962 rad` as well, so we repeat the process:

`(\pi)/(6)t=4.1962`

`t=(4.1962(6))/(\pi)`

`t=8.014 months`

So now we need to determine on which period of times the number of items sold exceed 77 hundred units so we build different intervals for us to test:

(1,3.98) (3.98,8.014) and (8,014, 13)

and find a test value for each of the intervals and test it.

(1,3.98) t=2

`S=56.9-40.7cos ((\pi)/(6)(2))`

S=36.55

this is less than 77 so this is not our answer.

(3.98,8.014) t=5

`S=56.9-40.7cos ((\pi)/(6)(5))`

S=92.15

this is more than 77 so this is our answer.

(8.014,13) t=10

`S=56.9-40.7cos ((\pi)/(6)(10))`

S=36.55

this is less than 77 so this is not our answer.

so, since our answer is the interval (3.98,8.014)

this means that between the months of march and august we will be sellin more than 7700 units.