Donna De Paul is raising money for the homeless. She discovers that each church group requires 2 hours of letter writing and 1 hour of follow-up, while for each labor union she needs 2 hours of letter writing and 3 hours of follow-up. Donna can raise $\$ 100$ from each church group and $\$ 175$ from each union local, and she has a maximum of 16 hours of letter writing and a maximum of 16 hours of follow-up available per month. Determine the most profitable mixture of groups she should contact and the most money she can raise in a month.
Let $\mathrm{x}_{1}$ be the number of church groups, and let $\mathrm{x}_{2}$ be the number of labor unions. What is the objective function?
\[
z=100 x_{1}+175 x_{2}
\]
(Do not include the $\$$ symbol in your answers.)
She should contact $\square$ church group(s) and $\square$ labor union(s), to obtain a maximum of $\$ \square$ in donations. (Simplify your answers.)
Answer
Final Answer: She should contact \(\boxed{4}\) church group(s) and \(\boxed{4}\) labor union(s), to obtain a maximum of \$ \(\boxed{1100}\) in donations.
Step 1 ：Let \(x_{1}\) be the number of church groups, and let \(x_{2}\) be the number of labor unions. The objective function is \(z=100 x_{1}+175 x_{2}\).
Step 2 ：We are trying to maximize the objective function \(z=100x_1+175x_2\) subject to the constraints \(2x_1+2x_2 \leq 16\) (letter writing time) and \(x_1+3x_2 \leq 16\) (follow-up time). We also have the non-negativity constraints \(x_1 \geq 0\) and \(x_2 \geq 0\).
Step 3 ：The optimal solution is obtained when Donna contacts 4 church groups and 4 labor unions. This will allow her to raise a maximum of $1100 in donations.
Step 4 ：Final Answer: She should contact \(\boxed{4}\) church group(s) and \(\boxed{4}\) labor union(s), to obtain a maximum of \$ \(\boxed{1100}\) in donations.
