one equation is for 40 <= x <= 80
the other equation for x > 80.
for x = 40 to 80, the equation is y = 16 * x.
it's a straight line equation and it will max out at x = 80.
for x > 80, the equation is y = (16 - .08 * (x - 80)) * (80 + (x - 80)).
this equation can be simplified to y = (16 - .08 * x + .08 * 80) * x.
simplify further to get y = (22.4 - .08 * x) * x.
simplify further to get y = 22.4 * x - .08 * x^2.
you have two equations.
y = 16 * x for 40 <= x <= 80
y = 22.4 * x - .08 * x^2 for x >= 80
when x = 80, the first equation gives you 16 * 80 = 1280 and the second equation gives you 22.4 * 80 - .08 * 80^2 = 1280.
in other words, they give you the same daily profit when x = 80.
you use the first equation for 40 <= x <= 80
you use the second equation for x > 80.
when x = 81, the daily profit will be 22.4 * 81 - .08 * 81^2 = 1289.52.
this is the same as (16 - .08 * 1) * (80 + 1) = 1289.52.
when x = 82, the daily profit will be 22.4 * 82 - .08 * 82^2 = 1298.88.
this is the same as (16 - .08 * 2) * (80 + 2) = 1298.88
the daily profit will keep rising until you hit 140 places.
when x = 140, 22.4 * 140 - .08 * 140^2 = 1568.
this is the same as (16 - .08 * 60) * (80 + 60) = 1568.
after 140 places, the daily profit starts declining.
at 141 places, the daily profit will be 22.4 * 141 - .08 * 141^2 = 1567.2.
it keeps going down from there.
i graphed the two equations.
the graphs looks like this.
the first graph is for 40 <= x <= 80
the second graph is for x >= 80
the third graph combine them into one graph.
the red equation is for 40 <= x <= 80
the blue equation is for x >= 80
both equations give you the same value at x = 80.
assuming i did this correctly, the maximum daily profit will be when 140 places are available and used.
the quadratic equation is y = 22.4 * x - .08 * x^2
reorder the terms in descending order of degree and you get y = -.08 * x^2 + 22.4 * x.
since this equation is in standard form, then:
a = 22.4 and b = -.08.
the maximum value will be when x = -b/(2a) which becomes x = -22.4 / -.16 = 140, as shown on the graph.