Question

Use the given cost table for the same product from two different companies to create alinear system. Then solve the system to determine when the cost of the product will be thesame and what the price will be.Two online retailers sell organic vanilla extract by the ounce using the following pricingchart.Vanilla Extract (oz)Chef Matec(n)Grocery Gourmetg(n)2$16.00$23.003$21.50$28.004$27.00$33.005$32.50$38.00din)n +n +g(n) =forounces of vanilla extract.Both Chef Mate and Grocery Gourmet charge $

Answer 1

According to the given table, for 2 ounces of vanilla extract, the cost is $16.00 at Chef Mate and $23.00 at Grocery Gourmet. For 4 ounces, the costs are $27.00 and $33.00, respectively.

Based on the given information, we can find the respective slopes and equations.

`\begin{gathered} m_{\text{Chef}}=(27-16)/(4-2)=(11)/(2) \\ m_{\text{Grocery}}=(33-23)/(4-2)=(10)/(2)=5 \end{gathered}`

Then, we use the point-slope formula to find the equations.

`\begin{gathered} y-y_1=m(x-x_1)_{} \\ y-27=(11)/(2)(x-4) \\ y=(11)/(2)x-(44)/(2)+27 \\ y=(11)/(2)x-22+27 \\ y=(11)/(2)x+5 \end{gathered}`

This equation represents c(n).

`c(n)=(11)/(2)n+5`

`\begin{gathered} y-y_1=m(x-x_1) \\ y-33=5(x-4) \\ y=5x-20+33 \\ y=5x+13 \end{gathered}`

This equation represents g(n).

`g(n)=5n+13`

To find the amount of vanilla extract where the cost is equal.

`\begin{gathered} c(n)=g(n) \\ (11)/(2)n+5=5n+13 \\ (11)/(2)n-5n=13-5 \\ (11n-10n)/(2)=8 \\ n=16 \end{gathered}`

We use this value to find the cost.

`c(16)=(11)/(2)\cdot16+5=88+5=93`

Hence, the price is $93.

Both Chef Mate and Grocery Gourmet charge $93 for 16 ounces of vanilla extract.