Question

which system below properly models the constraints as a system of inequalities? a \

$$\begin{cases} 5a + 25b \\geq 375 \\\\ a + b \\leq 35 \\\\ a \\geq 15 \\end{cases}$$

b \

$$\begin{cases} 5a + 25b \\geq 375 \\\\ a + b < 35 \\\\ b > 15 \\end{cases}$$

c \

$$\begin{cases} 5a + 25b \\geq 375 \\\\ a + b \\leq 35 \\\\ b \\geq 15 \\end{cases}$$

d \

$$\begin{cases} 5a + 25b \\geq 375 \\\\ a + b < 35 \\\\ a > 15 \\end{cases}$$

Explanation:

Step1: Analyze the first inequality

All options have \(5A + 25B\geq375\), so we focus on the other two inequalities.

Step2: Analyze the second inequality

Options A and C have \(A + B\leq35\), B and D have \(A + B\lt35\). Usually, in constraint problems (like resource or quantity limits), "at most" (≤) is more common than "less than" (<) as it includes the boundary. So likely A or C.

Step3: Analyze the third inequality

Option A has \(A\geq15\), Option C has \(B\geq15\). We need to determine which variable's lower bound is 15. Since the problem's context (even though not fully given) likely has a lower bound on \(B\) (as in typical problems where one variable has a minimum), and \(B\geq15\) in C, while A has \(A\geq15\) which may not fit. Also, the second inequality \(A + B\leq35\) (inclusive) is more appropriate than strict inequality. So C is the best fit.

Answer:

C. \(

$$\begin{cases}5A + 25B\geq375\\A + B\leq35\\B\geq15\end{cases}$$

\)