Question 21·Hard·Ratios, Rates, Proportional Relationships, and Units
A powdered sports-drink mix is prepared so that the finished drink has a mass ratio of 1 part powder to 16 parts water. The drink will be dispensed into 9 identical tanks, each of which holds 36 liters of the prepared drink. The density of the finished drink is approximately . The powder is sold only in 0.75-kilogram canisters that cost $12 each.
What is the minimum total cost of powder needed to fill all 9 tanks?
For mixture and ratio problems like this, move step by step with units: (1) find the total volume needed, (2) convert volume to mass using the given density, (3) apply the mass ratio by turning it into a fraction of the total (here, powder is of the total mass), and (4) convert the required powder mass into a count of packages, remembering that you must round up to a whole package before multiplying by the unit price. Keeping track of units at each step helps prevent common mistakes and speeds up checking your work.
Hints
Start with total volume and mass
First figure out how many liters of drink are needed in total for all 9 tanks, then use the density to convert that volume to a mass of drink.
Use the ratio to isolate the powder
The ratio is 1 part powder to 16 parts water. How many total parts is that? What fraction of the total mass is powder?
Connect powder mass to canisters and cost
Once you know the mass of powder, divide by per canister. If the result is not a whole number, what should you do? Then multiply the number of canisters by $12 to find the total cost.
Desmos Guide
Compute total mass of drink
In Desmos, type 1.05 * 9 * 36 to combine the density, number of tanks, and volume per tank. The output is the total mass (in kilograms) of the finished drink.
Find the mass of powder
In a new line, type (1/17) * (1.05 * 9 * 36) to multiply the total mass by . The result is the mass of powder needed (in kilograms).
Find the number of canisters
On another line, type ( (1/17) * (1.05 * 9 * 36) ) / 0.75. The output is the number of 0.75-kg canisters required; note that this value is not a whole number, so you must round up to the next integer.
Compute total cost
Finally, type roundUp( ( (1/17) * (1.05 * 9 * 36) ) / 0.75 ) * 12 if your Desmos has roundUp, or manually multiply the whole number of canisters you found by 12. The resulting value is the total cost in dollars; choose the answer choice that matches this value.
Step-by-step Explanation
Find the total volume of drink needed
There are 9 tanks, each holding 36 liters of drink.
Total volume:
Convert volume to mass using density
The density of the drink is , so mass = density × volume.
Compute it:
- Add:
So the total mass of the finished drink is .
Use the ratio to find the mass of powder
The mass ratio is 1 part powder to 16 parts water.
- That means there are total parts.
- Powder is of those parts, so powder is of the total mass.
So the mass of powder needed is
If you divide by , you get about of powder (a little more than kg).
Find how many canisters of powder are needed
Each canister has of powder.
Number of canisters needed:
is a little more than , so the exact result is about canisters.
You cannot buy a fraction of a canister, so you must buy 27 canisters to have enough powder.
Compute the total cost of powder
Each canister costs $12, and you need 27 canisters.
Total cost:
So the minimum total cost of powder needed to fill all 9 tanks is $324.