Mathematics, 22.05.2020 18:03, caden8233

1. An advertising company charges $40 per half-page advertisement and $75 per full-page advertisement. Michael has a budget of $800 to purchase 13 advertisements. (a) Define a variable for each unknown. Write a system of equations to represent the situation. (b) How many half-page advertisements does Michael purchase? Show your work. (c) How many full-page advertisements does Michael purchase? Show your work.

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(a) define a variable for each unknown and write a system of equations to represent the situation.
call x the number of half-page advertisements and y the number of full-page adverstisements
=>
40x + 75 y is the money charged, and given that he has a budget of $ 800
40x + 75y ≤ 800
the number of advertisements is 13, so, the other equation is:
x + y = 13
you coul also state the x and y cannot be negative, i.e. x ≥ 0 and y ≥ 0
so the complete system is:
40x + 75y ≤ 800
x + y = 13
x ≥ 0, y ≥ 0
(b) how many half-page advertisements does michael purchase? show your work.
in a graph draw the lines
40x + 75y = 800 > shade the are between this line, in the first quadrant.
x + y = 13
the solution is the on the part of the line x + y = 13 that is inside in the shaded region.
an addtional fact that you need to know is that the best solution (optimus) is one vertex of the solution area (this is one of the edges of the segment that we determined as possible solutions).
on of the edges is the point (0,13) and the other is the intercection point between the lines
40x + 75y = 800 and
x + y = 13.
find the solution of that system:
40 (13-y) + 75y = 800
520 - 40y + 75y = 800
35y = 800 - 520
35y = 280
y = 280/35
y = 8 = x = 13 - 8 = 5.
so the two possible solutions are
1) x = 5, y = 8
2) x = 13, y = 0 but with this solution he will not spend all the budget, because 40(13) +75(0) = 520.
so the solution is x = 5 and y = 8
that means that the number of half-page advertisements is 5
(c) how many full-page advertisements does michael purchase? show your work.
the work is above. there you got y = 8 which means that he purchases 8 full advertisements.
look that 40(5) + 75 (8) = 200 + 600 = 800.
which means that he spends all the budget when purchases 5 half-page advertisements and 8 full-page advertisements.
we also assured that this is the optimal solution.

answered: Guest

the range of a function is the range of y-values for which it is defined, or exists. given that we are only looking at the y-values, we can already discount the first two answers as they deal with the domain of the function (range of x-values).

now, what we need to do is find the smallest and highest y-values for which the function is defined. if you'd like, you could take a ruler and start at the very bottom of the graph, find the lowest point, then move the ruler higher and higher until you reach the highest point. doing so either in your head or with an actual ruler, we can see that the lowest y-value is -5 and the highest y-value is -1.

the question now remains as to whether they are included in the range (fourth answer) or excluded (third answer). the key here is to recognise whether they are marked with an open or a closed circle. since both (-2, -5) and (2, -1) are marked with a closed circle, they are points that exist on the graph and are thus included in the range (note that if they were marked with an open circle like (4, -2) then that would signify that there is a break in the graph at that point).

thus, the fourth answer, (y | -5 ≤ y ≤ -1), is correct.

also note that usually it is good practice to scan your eye inbetween the highest and lowest point for any breaks in the graph - we do have one here at (4, -2), however -2 is still included in the range because there is another point that is included in the graph with the same y-value, ie. point (1, -2).

i hope that but if you require any further explanation don't hesitate to comment below.