Problem 6 - Technical Mathematics with Calculus

Question

Treat the percents given in this exercise as exact numbers, and work to three significant digits. A certain chain saw requires a fuel mixture of \(5.5 \%\) oil and the remainder gasoline. How many liters of \(2.5 \%\) mixture and how many of \(9.0 \%\) mixture must be combined to produce 40.0 liters of \(5.5 \%\) mixture?

Step-by-Step Solution

Answer

To obtain 40.0 liters of 5.5% mixture, combine approximately 27.3 liters of 2.5% mixture and 12.7 liters of 9.0% mixture.

Step 1: Understanding the Problem

We need to find the amount of two mixtures that when combined will result in a new mixture with a certain percentage of oil. Specifically, we need to combine a mixture with 2.5% oil and a mixture with 9.0% oil to get 40.0 liters of a mixture that is 5.5% oil.

Step 2: Setting Up the Equation

Let the amount of 2.5% oil mixture be x liters and the amount of 9.0% oil mixture be y liters. According to the conditions we must have x + y = 40 (the total volume of the mixture). Additionally, the total amount of oil from both parts must equal 5.5% of the final mixture: 0.025x + 0.090y = 40 * 0.055.

Step 3: Solving the System of Equations

We have the following system of equations:\[\begin{align*}x + y &= 40,\ 0.025x + 0.090y &= 40 * 0.055.\end{align*}\]First we solve the first equation for y: \[ y = 40 - x \]. We then substitute this into the second equation and solve for x.

Step 4: Substitute and Solve for x

Substituting for y in the second equation gives: \[0.025x + 0.090(40 - x) = 40 * 0.055\]. We simplify and solve for x to find the liters of the 2.5% mixture.

Step 5: Find the value of y

Once we have x, we calculate y using the equation y = 40 - x.

Step 6: Check and Interpret the Answer

We check if the values of x and y are reasonable and if when we combine these amounts, we get the desired 5.5% oil mixture in 40 liters.

Key Concepts

Percent Concentration

Percent concentration is a way of expressing the amount of one substance (solute) present in a mixture with another substance (solvent), usually as a ratio in terms of volume or mass. In the context of solutions, percent concentration is typically denoted as a percentage, representing the volume or mass of the solute compared to the total volume or mass of the solution. For example, a \(5.5\%\) oil mixture indicates that for every 100 liters of the mixture, there are 5.5 liters of oil.

Understanding percent concentrations is vital when preparing solutions in various practical applications, such as mixing fuels or diluting chemicals. Getting the correct mixture proportions is essential to ensure the proper function of engines in the case of fuel or the desired reaction in chemical solutions. For instance, in the exercise given, the chainsaw requires a fuel mixture with a precise \(5.5\%\) oil concentration, necessitating a precise calculation to maintain engine health and efficiency.

When we mix different solutions, the resulting mixture's percent concentration can be calculated by considering the volumes and concentrations of the individual mixtures. The total volume of the final solution is the sum of the volumes of the individual mixtures, while the total amount of the solute (oil in this example) is the sum of the solutes in the individual mixtures. This principle allows us to create formulas for determining how much of each mixture to use.

Solving System of Equations

In mathematics, a system of equations is a collection of two or more equations with a common set of variables. Solving a system means finding the set of values for the variables that satisfy all the equations simultaneously. One common way to solve systems of linear equations is through the substitution or elimination method.

In our exercise, we are dealing with a system of linear equations where the variables represent the volumes of the oil mixtures we need to combine. The first equation \(x + y = 40\) represents the total volume constraint. The second equation \(0.025x + 0.090y = 40 \times 0.055\) expresses the percentage concentration of oil that the final mixture should have. To solve this system, we often isolate one variable in terms of the other using the first equation and then substitute into the second. This method simplifies the system to one variable that we can solve for directly.

Once we solve for one variable, the other variable's value is found by plugging back into one of the original equations. This process is a cornerstone of algebra and is widely used across numerous scientific and engineering disciplines. In our context, it ensures we can find the exact volumes of two mixtures to form a new mixture with the desired percent concentration.

Volume of Mixtures

Volume of mixtures refers to the space occupied by combining two or more substances. In the context of our exercise, we are mixing two different oil-gasoline mixtures to reach a desired volume and concentration of a new mixture. Total volume is a key component since it dictates the amount of each mixture required to reach the target mixture.

When we talk about volume in mixtures, it is essential to remember that volumes are additive. This means that when mixing two solutions, the volume of the resultant mixture is equal to the sum of the individual volumes, assuming there is no volume change due to the mixing process. We can use the concept of additivity to set up equations that help us find the amounts needed from each solution to get to a specific total volume and concentration.

In practical settings, this often involves measuring liquids in beakers or tanks and combining them while considering their respective volumes and concentrations. Deep understanding of how volumes combine is crucial for precision in creating solutions, and neglecting this can lead to incorrect concentrations, which, in applications like the fuel mixture in the exercise, could damage equipment or produce inefficient results.