Please help turned in today an advertising company charges $40 per half page advertisements and $75 per full page advertisement. Michael has a budget of $800 to purchase 13 advertisement.
(a) define a variable for each unknown.
write a system of equation to represent the situation.
(b) how many half page advertisements does Michael purchase?
show your work
(c) how many full page advertisement dose Michael purchase?
show your work

Step-by-step explanation:

For part a)

Call x the number of half-page advertisements and y the number of full-page advertisements

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 could 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

For part b)

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 additional fact that you need to know is that the best solution (optimum) 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 intersection point between the lines

40x + 75y = 800 and

x + y = 13.

Find the solution of that system by solving the two simultaneous equations by substitution:

40 (13-y) + 75y = 800

520 - 40y + 75y = 800

35y = 800 - 520

35y = 280

y = 280/35

y = 8 and for x

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

For part c)

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.  

I also attached a print screen image of the graphs where both the lines intersect is the solution it is confirmed that the lines intersect at (5,8) so that is our optimum solution.

the arc length is found by finding the circumference of the full circle (2xpixradius) and then dividing by 4 to find just one quarter of the circle (90 degrees).