savings = 100 * x + 10 * y
100 * x is the savings from replacing x light bulbs.
10 * y is the savings from insulating y square feet of exterior wall.
constraint functions are:
160x + 45y <= 4800
160 * x is the cost to replace x light bulbs.
45 * y is the cost to insulate y square feet of exterior wall.
x <= 80
y <= 1000
x is the total number of replaced light bulbs.
y is the total number of insulated square feet of exterior wall.
max of 80 light bulbs can be replaced.
max of 1000 square feet of exterior wall can be insulated.
to summarize:
objecting function:
s = 100 * x + 10 * y
constraint functions:
160 * x + 45 * y <= 48000
x <= 80
y <= 1000
using the desmos.com calculator, you would graph the opposite of the inequalities.
the area on the graph that is not shaded is the region of feasibility.
the corner points on the graph would contain the maximum savings.
you would evaluate the objective function at each of these corner points.
the corner point with the greatest value gives you your maximum savings.
each of the corner points needs to satisfy the constraint functions.
since your answer needs to be an integer, you would need to determine how to adjust to have the answer give you integer values and still provide the maximum savings and still satisfy all of the constraints.
maximum savings occur when 80 lightbulbs are replaced and 782.222 square feet of exterior wall are insulated.
here's the graph.
the maximum savings are at the coordinate point of (80,782.222)
this means x = 80 and y = 782.222.
the constraint equations need to be satisfied at that point.
x <= 80 is satisfied.
y <= 1000 is satisfied.
160x + 45y <= 48000 is satisfied because 160 * 80 + 45 * 782.222 = 47999.99.... which is less than or equal to 48000.
you can't round y up because then your cost would be greater than 48000, so you have to round it down to 782.
you maximum savings are now 100 * 80 + 10 * 782 = 15,820.
they are still greater than the savings of all the other options.
(0,1000) gave you a savings of 10,000
(80,0) gave you a savings of 8000
(18.75,1000) gave you a savings of 11,500.
(80,782.222) gave you a savings of 15,822.22.
the constraints at (18.75,1000) were equal to 48,000, but the number of light bulbs was not an integer.
you could go through and figure out the maximum savings when you made x an integer and still stay under the constraints, but it's not worth it because you would still not have savings greater than at (80,782.222).
let me know if you have any questions.
maximum savings at (80,782.222)
round to integer to get (80,782)