Answer:
- x + y ≥ 60
- 400x +1200y ≤ 68000
- y ≥ 3x
- x > 0, y > 0
- P = 1900x +14500y
1 radio ad and 63 TV ads will expose 915,400 people
Explanation:
You want various inequalities as described by the given advertising scenario.
Variables
Ordinarily, we might want to choose variables that remind us what they stand for. We could use r and t, for example, to represent the numbers of radio and TV ads. However, this problem statement defines the variables as x and y. We don't know what they are supposed to stand for, so we'll adopt the rule ...
- x is the number of radio ads
- y is the number of television ads
1. Total ads
The problem text tells you the plan is to buy at least 60 total ads. The total number of ads is (x+y), and the wording "at least" means "greater than or equal to." This inequality is ...
x + y ≥ 60
2. Ad cost
The total cost of the ads is the sum of costs of radio ads and television ads. The cost of ads is the number of ads times the cost of that kind of ad. The cost of all radio ads will be 400x; the cost of all TV ads will be 1200y. The problem text tells you the budget is $68000, so the total cost will be less than or equal to this amount. The inequality is ...
400x +1200y ≤ 68000
3. Ad ratio
The number of radio ads is x. Three times that number is 3x. The number of TV ads will be at least this amount, so we use the "greater than or equal to" inequality symbol:
y ≥ 3x
4. Other constraints
As with many problems of this type, we want the values of the variables to be positive. The inequality symbol is already given for you as "greater than". Positive values will be greater than zero.
x > 0 and y > 0
5. Objective function
The problem text tells you each radio ad reaches 1900 people. So, the number of people exposed is 1900x. Each TV ad will be seen by 14500 people, so the number of people exposed is 14500y. The total number of people exposed will be the sum of these numbers:
P = 1900x +14500y
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Additional comments
These inequalities are graphed in the attachment. Each one is graphed in reverse, so the solution space is left white, and the shaded areas are excluded. The dashed lines are part of the solution space.
You may notice we excluded x < 1, which is not the precise opposite of x ≥ 0. This is for a couple of reasons. (1) the smallest integer greater than 0 is 1, so x > 0 is effectively x ≥ 1 for integer values of x. (2) The vertex of the solution space that maximizes the objective function (people exposed) is (x, y) = (1, 63). By graphing x < 1, the best solution is on the corner of the solution space, as it should be.
Then the answer to "How many of each type of ad ..." is 1 radio and 63 TV.
Ordinarily, you would choose a the answer to the number of ads question by evaluating the objective function at the vertices of the solution space:
(1, 59), (1, 63), (19, 57), (15, 45)
When we draw the line representing the objective function, we see it is fairly flat, so it will have a maximum value at the vertex highest on the graph, (1, 63). The number of people exposed to the advertising is 915,400 with that mix of (radio, TV) ads. Note that this uses the entire ad budget.
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