Chapter 11.8, Problem 30AYU

Animal Nutrition Kevin's dog Amadeus likes two kinds of canned dog food. Gourmet Dog costs 40 cents a can and has 20 units of a vitamin complex; the calorie content is 75 calories. Chow Hound costs 32 cents a can and has 35 units of vitamins and 50 calories. Kevin likes Amadeus to have at least 1175 units of vitamins a month and at least 2375 calories during the same time period. Kevin has space to store only 60 cans of dog food at a time. How much of each kind of dog food should Kevin buy each month to minimize his cost?

To determine

To solve: The given linear programming problem.

Answer to Problem 30AYU

Solution:

15 numbers of Gourmet Dog cans and 25 numbers of Chow Hound cans are to be purchased to minimize the cost.

Explanation of Solution

Given:

• The costs of Gourmet Dog can and Chow Hound Can are 40 cents and 32 cents respectively.
• The vitamin content in Gourmet Dog can and Chow Hound Can are 20 units and 35 units respectively.
• The calories provided by each Gourmet Dog can and Chow Hound Can are 75 units and 50 units respectively.
• The dog requires at least 1175 units of vitamin and 2375 calories a month.
• Only 60 can of dog food can be stored at a time.

Calculation:

Begin by assigning symbols for the two variables.

$x$ be the number of Gourmet Dog cans.

$y$ be the number of Chow Hound cans.

If $C$ be the total cost of buying cans for a month,

$C=40x+32y$

The goal is to minimize $C$ subject to certain constraints on $x$ and $y$. Because $x$ and $y$ represents number of dog food cans, the only meaningful values of $x$ and $y$ are non-negative.

Therefore, $x\ge 0,y\ge 0$.

From the given data we get,

$20x+35y\ge 1175\Rightarrow 4x+7y\ge 235$

$75x+50y\ge 2375\Rightarrow 3x+2y\ge 95$

$x+y\le 60$

Therefore, the linear programming problem may be stated as,

Minimize, $C=40x+32y$.

Subject to,

$x\ge 0$

$y\ge 0$

$4x+7y\ge 235$

$3x+2y\ge 95$

$x+y\le 60$

The graph of the constraints is illustrated in the figure below.

The corner points are as follows:

Corner points are	Value of objective function
$\left(0,60\right)$	$1920$
$\left(0,48\right)$	$1536$
$\left(15,25\right)$	$1400$
$\left(60,0\right)$	$2400$
$\left(59,0\right)$	$2360$

Therefore, 15 number of Gourmet Dog cans and 25 number of Chow Hound cans are to be purchased to minimize the cost.