# help

• Dec 4th 2006, 07:02 AM
zasi
help
a musical group's new single is released weekly sales s (in thousands) increase steadily for a while and then decrease as given by the function s = -2<t-20> + 40 where t is the time (in weeks). what was the maximum number of singles sold in one week?
• Dec 4th 2006, 07:15 AM
Soltras
Quote:

Originally Posted by zasi
a musical group's new single is released weekly sales s (in thousands) increase steadily for a while and then decrease as given by the function s = -2<t-20> + 40 where t is the time (in weeks). what was the maximum number of singles sold in one week?

This function always decreases.
$\displaystyle s = -2(t - 20) + 40 = -2t + 80$.

This means week one, the group will sell 78,000 records.
Week two, it'll sell 76,000 records.
Week three, it'll sell 74,000 records.
etc

Clearly week one saw the highest sales.
(was that the correct formula? your description mentioned an initial rise. Are you sure it's not $\displaystyle -2t(t-20)+40$ or something?)
• Dec 4th 2006, 07:33 AM
zasi
help
the formula is base on absolute value of the function i.e.y = a[x - h] + k
• Dec 4th 2006, 07:48 AM
Soltras
All right, so the formula is $\displaystyle s = -2 \left| t - 20 \right| + 40$?

Want to maximize this.

Well there are two terms being added, $\displaystyle -2 \left| t - 20 \right|$ and $\displaystyle 40$. We want them both to be as large as possible. Obviously we can't do anything to the 40, so we just have to maximize $\displaystyle -2 \left| t - 20 \right|$.

Well this is never positive because it's -2 times an absolute value (which is never negative). So to maximize it, we just make it as "least" negative as possible -- that is, minimize the value of this term.
Setting to 0 should to the trick.

Set $\displaystyle 0 = -2 \left| t - 20 \right|$.
Then $\displaystyle 0 = \left| t - 20 \right|$.
Because it's equal to zero, we can drop the absolute value bars to get:
$\displaystyle 0 = t - 20$.

So $\displaystyle t = 20$.
This will maximize the function.

Substitute it into the original function:
$\displaystyle s(20) = -2 \left| (20) - 20 \right| + 40 = 40$.

So I claim that 40,000 records is most sold in a single week, and it occurs at week 20.
Verify for good measure: Try weeks 19 and 21.

$\displaystyle s(19) = -2 \left| (19) - 20 \right| + 40 = -2 (1) + 40 = 38$, and
$\displaystyle s(21) = -2 \left| (21) - 20 \right| + 40 = -2 (1) + 40 = 38$.

There. Kind of sloppy, but hopefully helpful.