Q. 30 - Precalculus Enhanced with Graphing Utilities

Question

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? 

Step-by-Step Solution

Answer

Kevin should buy $15$ cans of Gourment Dog and $25$ cans of Chow Hound each month to minimize his cost.

Step 1. Given Information

For Gourmet Dog the price is $40$ cents per can, has vitamin and calorie content of $20$ units and $75$ calories respectively.

For Chow Hound the price is $32$ cents per can, has vitamin and calorie content of $35$ units and $50$  calories respectively.

At least $1175$ units of vitamins and at least $2375$ calories must be taken per month.

The storage capacity is of at most $60$ cans.

Step 2. Form the inequalites

Let Kevin buys $x$ cans of Gourmet Dog and $y$ cans of Chow Hound.

He had a storage of only $60$ cans, so $x+y\le 60$.

The vitamin consumption from eating $x$ cans of Gourmet Dog and $y$ cans of Chow Dog is given as $20x+35y$ and it should be greater than or equal to $1175$ units. So $20x+35y\ge 1175$.

The calorie consumption from eating $x$ cans of Gourmet Dog and $y$ cans of Chow Dog is given as $75x+50y$ and it should be greater than or equal to $2375$. So, $75x+50y\ge 2375$.

Also number of cans cannot be non negative, so

$$\begin{gathered} x\ge 0 \\ y\ge 0 \end{gathered}$$

Step 3. Form the expression that needs to be minimized

The cost of one can of Gourmet Dog is $40$ cents and cost of one can of Chow Hound is $32$ cents. So the total cost of buying $x$ cans of Gourmet Dog and $y$ cans of Chow Hound is given by the expression $40x+32y$.

And the cost needs to be minimized. So the expression $C=40x+32y$ needs to be minimum.

Step 4. Graph the inequalities

The inequalities are graphed and the feasible region has been shaded as  

Step 5. Find the cost at each corner point

The feasible region is marked by five boundary points and they are $\left(0,47.5\right),\left(0,60\right),\left(60,0\right),\left(58.75,0\right),\left(15,25\right)$.

Now the cost at each point is found as

$\left(x,y\right)$	$C=40x+32y$
$\left(0,47.5\right)$	$40·0+32·47.5=1520$
$\left(0,60\right)$	$40·0+32·60=1920$
$\left(60,0\right)$	$40·60+32·0=2400$
$\left(58.75,0\right)$	$40·58.75+32·0=2350$
$\left(15,25\right)$	$40·15+32·25=1400$

So it can be seen that minimum value of the cost is $1400$ cents and it is at point $\left(15,25\right)$.

So Kevin should buy $15$ cans of Gourmet Dog and $25$ cans of Chow Hound.