A landscaping company is planting x maple trees and y pine trees in a park. The number of … — SAT

A landscaping company is planting $x$ maple trees and $y$ pine trees in a park. The number of trees of each type must satisfy the system of inequalities below:

\begin{aligned} 30x + 40y &\le 360 \\ x + y &\ge 10 \\ x &\ge 4 \end{aligned}

What is the maximum number of pine trees the company can plant?

Cevap: 6

Cevap

The maximum number of pine trees the company can plant is 6.

To find the maximum number of pine trees,

y

, we look at the boundary constraints. The constraint

x \ge 4

states that at least

4

maple trees must be planted. Since planting fewer maple trees leaves more of the budget for pine trees, we minimize

x

by setting

x = 4

. Substituting this value into the budget inequality

30x + 40y \le 360

gives

120 + 40y \le 360

. Solving for

y

yields

40y \le 240

, which simplifies to

y \le 6

. We then verify that the solution

(4, 6)

satisfies the total tree constraint

x + y \ge 10

, which it does since

4 + 6 = 10

. Thus, the maximum number of pine trees is 6.

Adım Adım Çözüm

Substitute the minimum possible value of

x

into the first inequality.

Since

x \ge 4

, the smallest possible value for

x

4

. Substituting

x = 4

into

30x + 40y \le 360

yields:

30(4) + 40y \le 360

120 + 40y \le 360

To maximize the value of

y

under the resource constraint, we must minimize the value of

x

Solve the inequality for

y

40y \le 240

y \le 6

This establishes that the maximum possible value for

y

based on the budget constraint is

6

Verify that

(4, 6)

satisfies all inequalities in the system.

Checking the second inequality:

x + y \ge 10 \implies 4 + 6 = 10 \ge 10

, which is true. The third inequality

x \ge 4 \implies 4 \ge 4

is also true.

A coordinate pair must satisfy all inequalities in the system to be a valid solution.

Anahtar Kavram

To find the maximum value of a variable in a system of inequalities with constraints, analyze the boundary lines and the intersection points of the feasible region.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru