.
A company installs solar panels in its premises to reduce its electricity cost.
The monthly savings on electricity in $, is modelled by S=200+50x-2x²,
where x is the number of months after installation.
(a) At what time will the savings on electricity stop increasing?
(b) Find the maximum savings.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The given quadratic function describes a parabola.
This parabola is opened downward, because the leading coefficient at x^2 is negative.
So, the parabola has a maximum at its vertex.
(a) the given quadratic function is increasing on the left of the vertex.
x-coordinate of the vertex is , where 'a' = -2 is the coefficient at x^2
and 'b' = 50 is the coefficient at x. So, x-coordinate of the vertex is
= = 12.5.
For the further analysis, take into account that 'x' in this problem is, factually, not a continuous time,
but a discrete counter of months. The value of 12.5 is exactly half-way between 12 and 13.
Since the parabola is symmetric about the vertex coordinate, we round the time 12.5 months to 12 months.
It means that the monthly saving on electricity stops increasing at the 12-th month, according to the given data.
(b) To find the maximum monthly saving, calculate the value of the given quadratic function at x = 12.
The maximum monthly saving will be 200 + 50*12 - 2*12^2 = 512 dollars.
Below is the table showing monthly savings for the first 15 months calculated using the given formula.
The table confirms that the monthly payment increases till the month 12 and stops increasing then.
It also confirms that the maximum monthly payment is $512.
month monthly
saving
------------------------
1 248
2 292
3 332
4 368
5 400
6 428
7 452
8 472
9 488
10 500
11 508
12 512
13 512
14 508
15 500
Solved.