A digital marketing firm allocates its monthly advertising budget between search engine ca… — SAT

A digital marketing firm allocates its monthly advertising budget between search engine campaigns and social media campaigns. Let $x$ represent the amount, in thousands of dollars, spent on search engine campaigns, and let $y$ represent the amount, in thousands of dollars, spent on social media campaigns. The system of inequalities below represents the firm's monthly constraints:

\begin{aligned} x &\ge 2y \\ x + y &\le 25 \\ 3x + 4y &\le 80 \\ y &\ge 3 \end{aligned}

Based on these constraints, what is the maximum possible amount, in thousands of dollars, the firm can spend on social media campaigns?

Cevap: 8

Cevap

To find the maximum possible value of

y

, we analyze the boundaries of the feasible region. The boundary lines of the system are

x = 2y

x + y = 25

3x + 4y = 80

, and

y = 3

. We can find the upper limit of

y

by combining the inequalities

x \ge 2y

and

3x + 4y \le 80

. Multiplying the first inequality by

3

gives

3x \ge 6y

. Adding

4y

to both sides yields

3x + 4y \ge 10y

. Because

3x + 4y \le 80

, it follows that

10y \le 80

, which simplifies to

y \le 8

. We verify that the point

(16, 8)

satisfies the remaining inequalities:

16 + 8 = 24 \le 25

and

8 \ge 3

, which are both true. Thus, the maximum possible value of

y

8

Adım Adım Çözüm

Relate the variables using the constraints to establish an upper bound for

y

Since

x \ge 2y

, multiplying both sides by

3

gives

3x \ge 6y

This allows us to express the

3x

term in the cost inequality in terms of

y

to determine the maximum boundary.

Substitute

3x \ge 6y

into the inequality

3x + 4y \le 80

We get

6y + 4y \le 3x + 4y \le 80

, which simplifies to

10y \le 80

This establishes a direct upper limit for

y

based on the intersection of the two active boundary lines.

Solve the inequality

10y \le 80

for

y

y \le 8

This determines that the maximum possible value for

y

under these constraints is

8

Verify that the point corresponding to

y = 8

satisfies all other constraints in the system.

When

y = 8

, the boundary

x = 2y

gives

x = 16

. Checking

(16, 8)

against all inequalities:
-

16 \ge 2(8) \implies 16 \ge 16

(True)
-

16 + 8 \le 25 \implies 24 \le 25

(True)
-

3(16) + 4(8) \le 80 \implies 80 \le 80

(True)
-

8 \ge 3

(True)

We must verify that the optimal vertex lies within the feasible region defined by all four inequalities.

Anahtar Kavram

Maximizing a coordinate value within a bounded feasible region defined by a system of linear inequalities.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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