Chapter 8.3, Problem 19E

a.

To determine

To write:

An equation describing the possible amounts of canned and dry food that you can feed your beagle each day.

Answer to Problem 19E

Our required equation would be $40x+100y=800$ .

Explanation of Solution

Given:

Your beagle is allowed to eat 800 calories of food each day. You buy canned food containing 40 calories per ounce and dry food containing 100 calories per ounce.

Calculation:

Let x represent ounces of canned food and y represent ounces of dry food.

We have been given that each ounce of canned food contains 40 calories, so calories in x ounces of canned food would be $40x$ .

We are also told that each ounce of dry food contains 100 calories, so calories in y ounces of dry food would be $100y$ .

Since your beagle is allowed to eat 800 calories each day, so the total calories consumed from eating x ounces of canned food and y pounds of dry food should be equal to 800.

We can represent this information in an equation as:

  $40x+100y=800$

Therefore, our required equation would be $40x+100y=800$ .

b.

To determine

To graph:

The equation from part (a) using the intercepts.

Answer to Problem 19E

The graph of the equation $40x+100y=800$ would look like:

Explanation of Solution

Given:

Your beagle is allowed to eat 800 calories of food each day. You buy canned food containing 40 calories per ounce and dry food containing 100 calories per ounce.

Calculation:

To find x -intercept, we will substitute $y=0$ in our equation as:

  $\begin{array}{l}40x+100y=800\left(\text{Given equation}\right)\\ 40x+100(0)=800\left(\text{Substituting }y=0\right)\\ 40x=800\\ x=20\end{array}$

To find y -intercept, we will substitute $x=0$ in our equation as:

  $\begin{array}{l}40x+100y=800\left(\text{Given equation}\right)\\ 40(0)+100y=800\left(\text{Substituting }x=0\right)\\ 100y=800\\ y=8\end{array}$

Upon graphing our given equation, we will get our required graph as shown below:

c.

To determine

To give:

Three possible combinations of canned and dry food that you can feed your beagle.

Answer to Problem 19E

Three possible combinations would be:

1. (5,6)

2. (10,4)

3. (15,2)

Explanation of Solution

Given:

Your beagle is allowed to eat 800 calories of food each day. You buy canned food containing 40 calories per ounce and dry food containing 100 calories per ounce.

Calculation:

To find some possible combinations, we will substitute some values of xin our given equation as shown below:

  $\begin{array}{l}40x+100y=800\left(\text{Given equation}\right)\\ 40(5)+100y=800\left(\text{Substituting }x=5\right)\\ 200+100y=800\\ 100y=600\left(\text{Subtracting 200 from both sides}\right)\\ y=6\left(\text{Dividing both sides by 100}\right)\end{array}$

  $\begin{array}{l}40x+100y=800\left(\text{Given equation}\right)\\ 40(10)+100y=800\left(\text{Substituting }x=10\right)\\ 400+100y=800\\ 100y=400\left(\text{Subtracting 400 from both sides}\right)\\ y=4\left(\text{Dividing both sides by 100}\right)\end{array}$

  $\begin{array}{l}40x+100y=800\left(\text{Given equation}\right)\\ 40(15)+100y=800\left(\text{Substituting }x=15\right)\\ 600+100y=800\\ 100y=200\left(\text{Subtracting 600 from both sides}\right)\\ y=2\left(\text{Dividing both sides by 100}\right)\end{array}$

Therefore, the three possible combinations would be (5,6), (10,4) and (15,2).