Chapter 2.7, Problem 12E

(a)

To determine

To find: The inequality to represent the ounces of protein required.

Answer to Problem 12E

The inequality is $0.84x+0.56y\ge 1.54$ .

Explanation of Solution

Given:

The given diet that includes at least 1.54 ounces of protein. Each cup of Good start puppy food contains 0.84 ounce of protein. Each cup of Sirius puppy food contains 0.56 ounce of protein.

The inequality that represent the ounces of protein required as below.

  $0.84x+0.56y\ge 1.54$ .

Here the amount of protein given to each cup of good start puppy food is $0.84x$ and the amount of protein given to each cup of Sirius puppy food is $0.56y$ .

Therefore, the inequality is $0.84x+0.56y\ge 1.54$ .

(b)

To determine

To find: The logarithm model from the given data.

Answer to Problem 12E

The inequality is $0.21x+0.49y\ge 0.56$ .

Explanation of Solution

Given:

The given diet that includes at least 0.56 ounces of fat. Each cup of Good start puppy food contains 0.21 ounce of fat. Each cup of Sirius puppy food contains 0.49 ounce of fat.

The inequality that represent the ounces of fat required as below.

  $0.21x+0.49y\ge 0.56$ .

Here the amount of fat given to each cup of good start puppy food is $0.21x$ and the amount of fat given to each cup of Sirius puppy food is $0.49y$ .

Therefore, the inequality is $0.21x+0.49y\ge 0.56$ .

(c)

To determine

To find: The graph the system of inequalities.

Answer to Problem 12E

The graph is given below.

Explanation of Solution

The graph of inequality is given below:

  

In order to check whether the shaded part contains origin or not check the following.

  $\begin{array}{l}0.84\left(0\right)+0.56\left(0\right)\ge 1.54\left[\text{false}\right]\\ 0.21\left(0\right)+0.49\left(0\right)\ge 0.56\left[\text{false}\right]\end{array}$

The shaded region does not contain origin.

Therefore the graph is given above.

(d)

To determine

To find: The function to represent the daily cost of puppy food.

Answer to Problem 12E

The function to represent the daily cost of puppy food is $c\left(x,y\right)=36x+22y$ .

Explanation of Solution

The cost of Good start puppy food is 36$ per cup.

The cost of Sirius puppy food is 22$ per cup.

So the function to represent the daily cost of puppy food is given as:

  $c\left(x,y\right)=36x+22y$

Here $x$ is the number of cups used for Good Start puppy.

And $y$ is the number of cups used for Sirius puppy.

Therefore the function to represent the daily cost of puppy food is $c\left(x,y\right)=36x+22y$ .

(e)

To determine

To find: The number of cups of each type of puppy food should be used in order to minimise the cost.

Answer to Problem 12E

The number of cups of each type of puppy:1.5 cups of Good Start and $0.5$ cups of Sirius.

Explanation of Solution

In order to minimise the cost, test each point in the minimizing function, that are shown on the boundary of the graph.

  $\begin{array}{c}c\left(0,1.83\right)=36\left(0\right)+22\left(1.83\right)\\ \approx 40\\ c\left(1.5,0.5\right)=36\left(1.5\right)+22\left(0.5\right)\\ \approx 65\end{array}$

Further simplified as:

  $\begin{array}{c}c\left(2.67,0\right)=36\left(2.67\right)+22\left(0\right)\\ \approx 96\end{array}$

Although the values of the function at $\left(0,1.83\right)$ is the minimum but here $x=0$ so it means that one of foods is not going to be included in the mixture.

Therefore there is 1.5 cups of Good Start and $0.5$ cups of Sirius.

(f)

To determine

To find: The minimum cost.

Answer to Problem 12E

The minimum cost is $65$ .

Explanation of Solution

Calculation:

The minimum cost at $\left(0,1.83\right)$ is calculated as:

  $\begin{array}{c}c\left(1.5,0.5\right)=36\left(1.5\right)+22\left(0.5\right)\\ =65\end{array}$

Therefore the minimum cost is $65$ .