Question 42·Hard·Ratios, Rates, Proportional Relationships, and Units
A laboratory needs to prepare 560 liters of an alcohol solution that is at least alcohol by volume by mixing together a alcohol solution with a alcohol solution.
The solution is delivered only in sealed 8-liter barrels, and the laboratory must use a whole number of these barrels (barrels cannot be partially emptied or refilled).
What is the minimum number of 8-liter barrels of the solution that the laboratory must use?
For mixture problems with a target percentage, let a variable represent the amount of the stronger (or weaker) solution, write an equation or inequality for total pure substance (here, pure alcohol), and set it equal to or bounded by the target percentage times the total volume. Solve for the amount of the stronger solution, then carefully convert units (liters to barrels, hours to minutes, etc.) and pay attention to wording like "at least" or "at most," which often means you must round up or down to a whole item (such as barrels) and then check that this smallest integer still meets the condition.
Hints
Represent the amount of 45% solution algebraically
Let a variable stand for how many liters of the solution are used. In terms of this variable, how many liters of the solution will there be if the total must be 560 liters?
Use an inequality for "at least 32%"
Write an expression for the total liters of pure alcohol from both solutions, and set it greater than or equal to , since the mixture must be at least alcohol.
Solve, then think about barrels
After you solve the inequality for the liters of solution, divide by 8 to get the number of barrels. If this number is not an integer, what should you do to make sure the final mixture still has at least alcohol?
Check your candidate number
Try your candidate number of barrels: compute how many liters of each solution you would have and what percent alcohol the mixture would be. Does it reach and is it the smallest whole number that does?
Desmos Guide
Model the mixture percent as a function of barrels
Let be the number of 8-liter barrels of the solution. In Desmos, enter the function
This gives the alcohol fraction of the mixture for any number of barrels .
Find where the mixture reaches 32% and interpret
In Desmos, also enter the horizontal line . Find the -value where intersects ; this is the cutoff between too weak and strong enough. Since must be a whole number of barrels and the mixture must be at least , round this -value up to the next integer—that integer is the required number of barrels.
Step-by-step Explanation
Define the variable and volumes of each solution
Let be the number of liters of the alcohol solution used.
- Total mixture needed: liters.
- Liters of solution: .
- Liters of solution: .
Later, we will convert into a number of 8-liter barrels.
Write an inequality for the alcohol content
Total pure alcohol in the mixture comes from both solutions:
- From the solution: liters of alcohol.
- From the solution: liters of alcohol.
The mixture must be at least alcohol by volume, so the total alcohol must be at least of 560 liters:
Solve the inequality for the liters of 45% solution
Simplify and solve the inequality:
So the laboratory must use at least liters of the solution.
Convert liters to barrels and enforce whole barrels
Each barrel of the solution is 8 liters, so the number of barrels must satisfy
Thus
The lab can only use a whole number of barrels and must meet or exceed the requirement, so we must round up to the next whole number of barrels: 34 barrels.