Question

A young superhero mixes up a batch of goo they can use to fight crime. The recipe for the goorequires Distilled Water (which costs $2/gallon), Diet Cola (which costs $4/Gallon), and DilutedAlcohol (which costs $1/gallon). The batch of goo contains 8 gallons of these mixed ingredients andcosts $18. Furthermore, the batch of goo uses the same amount of Water and Cola. What is therecipe for the goo (i.e. how many gallons of Water, Cola, and Alcohol are used)?As a side note, it is possible to use a fraction of a gallon of an ingredient.а8a) Write out the system of equations you would use to answer the question.8b) What is the answer? How many gallons of each ingredient were used? Please write youranswer as a complete sentence with appropriate units.

Answer 1

Given

First, let's summarize the costs of each ingredient

Distilled Water: $2/gallon

Diet cola: $4/gallon

Diluted alchohol: $1/gallon

Total = 8 gallons

Cost = $18

Batch of goo uses the same amount of Water an Cola

Procedure

Let's formulate a system of equations to define the problem we have.

w = gallons of water

c = gallons of cola

a = gallons of alcohol

The first equation we can create is using the total amount of gallons created.

`w+c+a=8`

Now we can create an equation associated with costs and total cost

`2w+4c+a=18`

As you can see in our system we have 3 unknowns so we need a third equation.

For this we use the information that we need the same amount of Cola and Water.

`w=c`

8a) Let's write this as a system of equations

`\begin{gathered} w+c+a=8 \\ 2w+4c+a=18 \\ w=c \end{gathered}`

8b) Now let's proceed to solve the system of equations

`\begin{gathered} w=c,\text{ therefore } \\ w+w+a=8 \\ 2w+4w+a=18 \\ \\ 2w+a=8\text{ (1)} \\ 6w+a=18\text{ (2)} \\ \text{ using 1: } \\ a=8-2w,\text{ therefore} \\ 6w+8-2w=18 \\ 4w=18-8 \\ 4w=10 \\ w=10/4 \\ w=2.5 \end{gathered}`

As we know w = c, therefore c = 2.5. Now, for a

`\begin{gathered} a=8-2\cdot2.5 \\ a=3.0\text{ } \end{gathered}`

Water = 2.5 gallons

Cola = 2.5 gallons

Alcohol = 3.0 galons