• Sep 21st 2007, 05:30 PM
t-81
Please help!!! This assignment is due tomorrow and I can't seem to figure it out. If you can solve this, I need to see the work so I can figure out where I'm going wrong.

Here's the problem:

1. The Northwest Flower Company owns a greenhouse, which furnishes roses and carnations to florists in Oregon, Washington, and Idaho. The greenhouse can grow any combination of the two flowers. They sell the flowers in "bunches" with 25 blooms to a bunch. They have up to 10,000 square feet available for planting this year. Each bunch of roses takes about 4 square feet and each bunch of carnations about 5 square feet. Special fertilizer is required for flowers; roses need 5 pounds and carnations 2 pounds. The availability of the fertilize-pr is limited to 5000 pounds. Sales commitments require the company to grow at least 500 bunches of roses. Profit contributions are $6 per bunch of roses and$8 per bunch of carnations. What is the optimum profit and planting for this situation?
• Sep 21st 2007, 07:11 PM
Soroban
Hello, t-81!

Quote:

The Northwest Flower Company owns a greenhouse, which furnishes roses
and carnations to florists in Oregon, Washington, and Idaho.
The greenhouse can grow any combination of the two flowers.
They sell the flowers in "bunches" with 25 blooms to a bunch.

They have up to 10,000 square feet available for planting this year.
Each bunch of roses takes about 4 ft² and each bunch of carnations about 5 ft².

Special fertilizer is required for flowers; roses need 5 pounds and carnations 2 pounds.
The availability of the fertilizer is limited to 5000 pounds.

Sales commitments require the company to grow at least 500 bunches of roses.

Profit contributions are $6 per bunch of roses and$8 per bunch of carnations.

What is the optimum profit and planting for this situation?

Organize the given data . . .

$\begin{array}{cccccccc} & | & \text{Space} & | & \text{Fert.} & | & \text{Profit} & | \\ \hline
\text{Roses} & | & 4 & | & 5 & | & 6 & | \\ \hline
\text{Carnations} & | & 5 & | & 2 & | & 8 & | \\ \hline
\text{Available} & | & 10,000 & | & 5,000 & | & & \end{array}$

Let $x$ = number of bunches of roses: . ${\color{blue}x \,\geq \,0}$
Let $y$ = number of bunches of carnations: . ${\color{blue} y \,\geq \,0}$

Each bunch of roses takes 4 ft² of space.
. . $x$ bunches take $4x$ ft².
Each bunch of carnations takes 5 ft² of space.
. . $y$ bunches take $5y$ ft².
There are 10,000 ft² available.
. . Hence: . ${\color{blue}4x + 5y \:\leq\:10,000}$

Each bunch of roses requires 5 lbs of fertilizer.
. . $x$ bunches of roses require $5x$ lbs of fertilizer.
Each bunch of carnations requires 2 lbs of fertilizer.
. . $y$ bunches of carnations require $2y$ lbs of fertilizer.
There are 5,000 lbs of fertilizer available.
. . Hence: . ${\color{blue}5x + 2y \:\leq \:5,000}$

There must be at least 500 bunches of roses: . ${\color{blue}x \:\geq\:500}$

Graph and shade the five inequalities.
Determine the vertices $(x,y)$ of the critical region.
Then test the vertices in the profit function: . $P \:=\:6x + 8y$
. . to see which vertex produces maximum profit.