Natural Science Using the same assumptions about octane ratings as in Example 1, solve the...

Natural Science Using the same assumptions about octane ratings as in Example 1, solve the following problem.

How many liters of 94-octane gasoline should be mixed with 200 liters of 99-octane gasoline to get a mixture that is 97-octane gasoline?

Example 1

Business An oil company distributor needs to fill orders for 89-octane gas, but has only 87- and 93-octane gas on hand. How much of each type should be mixed together to produce 54,000 gallons of 89-octane gas?

Solution

Step 1 We must find how much 87-octane gas and how much 93-octane gas are needed for the 54,000 gallon mixture.

Step 2 Let x be the amount of 87-octane gas.

Step 3 Then 54,000 − x is the amount of 93-octane gas.

Step 4 The octane rating of a gasoline indicates that it has the same antiknock qualities as a standard fuel made of heptane and isooctane. The octane rating is the percentage of isooctane in the standard fuel.* So we assume that standard fuels are being mixed. We can summarize the relevant information in a chart.

Step 5 The amount of isooctane satisfies this equation:

Step 6 To solve the equation, use the distributive property on the left side and multiply out the right side:

Step 7 So the distributor should mix 36.000 gallons of 87-octane gas with 54,000 − 36.000 = 18,000 gallons of 93-octane gas. Then the amount of isooctane in the mixture is

Hence, the octane rating of the mixture is , as required.

*This is one of several possible ways of determining octane ratings.