exposure would be number of spots * the number of viewers that watch each spot.
if you let x equal the number of spots for family guy and y equal the number of spots for american dad,then your total exposure would be:
1.3 million * x + .9 million * y.
that's what you want to maximize.
that's your objective function.
note that the implicit assumption is that number of spots is the same as number of episodes.
in other words, one spot per episode is assumed.
now to the constraints.
first of all, x and y have to be greater than equal to 0.
your formula become3s:
x >= 0
y >= 0
next you want the number of spots on family guy to be greater than 75% of the overall spots.
your formula becomes:
x >= (x + y)
next you want to air at least 30 episodes / spots total.
your formula becomes:
x + y >= 30
next your budget has to be less than or equal to 70,000.
since each spot on family guy costs 2000 and each on american dad costs 3000, your formula becomes:
2000 * x + 3000 * y <= 70,000
in summary, what you have are:
objective function to maximize:
1.3 million * x + .9 million * y
constraint functions:
x >= 0
y >= 0
x >= .75 * (x + y)
x + y >= 30
2000x + 3000y <= 70,000
using www.desmos.co/calculator, the easiest way to see the region of feasibility is to graph the opposite inequalities.
the region of the graph that is not shaded is your area of feasibility.
the corner points of that region are where the max/min values lie.
you would graph:
x <= 0
y <= 0
x <=.75 * (x + y)
x + y <= 30
2000x + 3000y >= 70,000.
your graph would look like this:
your corner points and the value of your objective function are:
(23.333, 7.778) = 37.331 million viewers.
(22.5, 7.5) = 36 million viewers.
(30, 0) = 39 million viewers.
(35, 0) = 45.5 million viewers.
you will maximize your exposure when you put 35 spots on family guy and 0 spots on american dad.
all the constraints have to be met.
when you put 35 spots on family guy and 0 spots on american dad, all your constraints are satisfied as shown below.
x >= 0
satisfied because x = 35
y >= 0
satisfied because y = 0
x >= .75 * (x + y)
satisfied because x = 35 and y = 0
x + y >= 30
satisfied because x = 35 and y = 0
2000x + 3000y <= 70,000
satisfied because 2000 * 35 + 0 * 3000 = 70,000
note that you did not have any requirements to put any spots on american dad.
you get maximum exposure for your money without putting any spots on american dad.
if you wanted to have some spots on american dad, you would have had to add that as an additional constraint, such as y >= 5.
in that case, using www.desmos.com/calculator, you would add the additional inequality of y <= 5 and your region of feasibility would change along with the associated corner points.
since at least some of your corner points are changed, then it is possible that your maximum value for the objective function will change as well.
without getting into the calculations, your graph would then look like this:
you can see that the region of feasibility (the unshaded region) is now smaller because the value of y had to be >= 5 rather than >= 0.