Q. 79 - Precalculus Enhanced with Graphing Utilities

Question

Prices of Fast Food 

One group of customers bought $8$ deluxe hamburgers, $6$ orders of large fries, and $6$ large colas for $\backslash (26.10$. A second group ordered $10$ deluxe hamburgers, $6$ large fries, and $8$ large colas and paid $\backslash )31.60$. Is there sufficient information to determine the price of each food item? If not, construct a table showing the various possibilities. Assume that the hamburgers cost between $\backslash (1.75$ and $\backslash )2.25$, the fries between $\backslash (0.75$ and $\backslash )1.00$, and the colas between $\backslash (0.60$ and $\backslash )0.90$. 

Step-by-Step Solution

Answer

There is not sufficient information to determine the price of each food. The required table shows various possibilities:

Step 1. Given Information

Cost of $8$ deluxe hamburgers, $6$ orders of large fries, and $6$ large colas $=\$26.10$.

Cost of $10$ deluxe hamburgers, $6$ orders of large fries, and $8$ large colas $=\$31.60$.

Range of cost for hamburgers lies between $\$1.75$ and $\$2.25$.

Range of cost for the fries lies between $\$0.75$ and $\$1.00$ .

Range of cost for the colas lies between $\$0.60$ and $\$0.90$.

Step 2. Using substitution and elimination method

Let $x$ be the cost of hamburgers, $y$ be the cost of fries and $z$ be the cost for colas.

The required equations are:

$$\begin{gathered} 8x+6y+6z=26.10........\left(i\right) \\ 10x+6y+8z=31.60........\left(ii\right) \end{gathered}$$

Multiplying equation $\left(i\right)$ by $-\frac{1}{8}$ and equation $\left(ii\right)$ by $\frac{1}{10}$, we get:

$$\begin{gathered} -x-0.75y-0.75z=-3.2625.........\left(iii\right) \\ x+0.6y+0.8z=3.16.........\left(iv\right) \end{gathered}$$

Adding equations $\left(iii\right)$ and $\left(iv\right)$ we get,

$$\begin{gathered} -0.15y+0.05z=-0.1025 \\ \Rightarrow -0.15y=-0.1025-0.05z \\ \Rightarrow y=\frac{0.1025}{0.15}+\frac{0.05z}{0.15} \\ \Rightarrow y=0.683+0.33z \end{gathered}$$

Substituting the value of $y$ in equation $\left(iv\right)$ we get,

$$\begin{gathered} x+0.6(0.683+0.33z)+0.8z=3.16 \\ \Rightarrow x+0.41+0.2z+0.8z=3.16 \\ \Rightarrow x+z=3.16-0.41 \\ \Rightarrow x=2.75-z \end{gathered}$$

Step 3. Creating the table

We determine the price of various items by taking different values of $z$ between $\$0.60$ and $\$0.90$.

The table shows the various possibilities