Answer :
there seem to be some typing mistake in question. so i do different cases:
case I : radio ads cost $15 each
Follow procedure explained for case II.
Constraints are: 10 x + 3 y <= 1800 and 5 x <= 3y
Maximization function: Reach = 8000 x + 6000 y
We get a triangle with O(0,0) , A(120,200) and B(180,0).
Maximum Reach is at A.
Maximum reach : 2.16 million
=============================================
Case II : radio ads cost $150 each.
Let us x newspaper ads are given and y radio ads are given.
The Spend on newspaper ads = x * $50
The spend on radio ads = y * $150
1st constraint :
x * $50 + y * $150 <= $9000
x + 3 y <= 180 ---- constraint 1
2nd constraint
The spend on Newspaper ads <= 2 * the Spend on radio ads
x * $50 <= 2 * y * $150
x <= 6 y or x - 6y <= 0 ------------- constraint 2
If we plot graphs of the two straight lines for the two constraints
they meet at A (120,20).
We also find x value for y = 0 from constraint 1. x = 180
We have a triangle with O(0,0) , B(180,0)
The maximization FUNCTION is Number of people the ads reach.
Reach = 8000 x + 6000 y
Find value of Reach at O, A and B. Choose the one with MAXIMUM reach.
Here it is B(180,0) Reach is 1.44 million
So X = 180 Y = 0
==============================
case III :
Constraint : total money spent is $9000. Money spent on newspaper ads is at most Half compared money spent on radio ads.
x * 50 <= 1/2 y * 150 => 2 x < 3y
and x*50 + y*150 <= 9000 => x + 3 y <= 180
solving these constraints , we get x = 60 and y = 40 for maximum reach
newspaper ads = 60 radio ads = 40
Maximum reach is 0.720 million people
==================
case I : radio ads cost $15 each
Follow procedure explained for case II.
Constraints are: 10 x + 3 y <= 1800 and 5 x <= 3y
Maximization function: Reach = 8000 x + 6000 y
We get a triangle with O(0,0) , A(120,200) and B(180,0).
Maximum Reach is at A.
Maximum reach : 2.16 million
=============================================
Case II : radio ads cost $150 each.
Let us x newspaper ads are given and y radio ads are given.
The Spend on newspaper ads = x * $50
The spend on radio ads = y * $150
1st constraint :
x * $50 + y * $150 <= $9000
x + 3 y <= 180 ---- constraint 1
2nd constraint
The spend on Newspaper ads <= 2 * the Spend on radio ads
x * $50 <= 2 * y * $150
x <= 6 y or x - 6y <= 0 ------------- constraint 2
If we plot graphs of the two straight lines for the two constraints
they meet at A (120,20).
We also find x value for y = 0 from constraint 1. x = 180
We have a triangle with O(0,0) , B(180,0)
The maximization FUNCTION is Number of people the ads reach.
Reach = 8000 x + 6000 y
Find value of Reach at O, A and B. Choose the one with MAXIMUM reach.
Here it is B(180,0) Reach is 1.44 million
So X = 180 Y = 0
==============================
case III :
Constraint : total money spent is $9000. Money spent on newspaper ads is at most Half compared money spent on radio ads.
x * 50 <= 1/2 y * 150 => 2 x < 3y
and x*50 + y*150 <= 9000 => x + 3 y <= 180
solving these constraints , we get x = 60 and y = 40 for maximum reach
newspaper ads = 60 radio ads = 40
Maximum reach is 0.720 million people
==================