Question

$$ 0.27(a+b)=0.15 a+0.35 b $$ An athletic trainer is attempting to produce a carbohydrate-electrolyte solution that is at \(27 \%\) carbohydrates by mass, which is the maximum amount of saturation allowed by her league. A supply company provides solutions that are at \(15 \%\) and \(35 \%\) carbohydrates by mass, respectively. Based on the equation above, if the trainer uses 10 quarts of the \(15 \%\) solution, how many quarts of the \(35 \%\) solution will she need? A) 180 B) 90 C) 30 D) 15 $$ f(x)=(x-b)^2-4 $$

Short answer

The trainer needs 15 quarts of the 35% solution.

Step-by-step solution

Substitute for a in the equation

From the problem, we know that the trainer is using 10 quarts of the 15% solution. This corresponds to \( a \) in the equation. So we replace \( a \) with 10: \( 0.27(10+b)=0.15*10+0.35b \).

Simplify the equation

Distribute the 0.27 across the brackets, and simplify the right side to get: \( 2.7+0.27b=1.5+0.35b \).

Rearrange the equation for b

To find \( b \), move all terms involving \( b \) to one side of the equation and constants to the other to get: \( 0.08b = 1.2 \).

Solve for b

Finally, divide both sides by 0.08 to find \( b \): \( b = 15 \). So the trainer needs 15 quarts of the 35% solution.

Evaluate f(x) for a given x

To evaluate the function \( f(x) = (x-b)^2 - 4 \) for a given value of \( x \), substitute that value into the equation and simplify. However, since we're not given a specific value of \( x \) to substitute, this can't be done at this point.

Key concepts

Understanding Percentage Calculations

Percentage calculations are an essential mathematical concept, used widely in various fields such as finance, data analysis, and scientific research. They help us understand proportions and compare values. In its essence, a percentage is a way of expressing a number as a fraction of 100. This can be represented as:

• If you have a number, say 40, and you need to find what percentage it is of 200, you would use the formula: \( \frac{40}{200} \times 100 = 20\% \)
• Similarly, to find the actual number from a percentage, rearrange the formula: \( \text{Number} = \frac{\text{Percentage}}{100} \times \text{Total} \)

In the context of mixture problems like the one given, these calculations help determine how much of each solution contributes to the final mixture percentage. The athletic trainer example involves two types of solutions, each with a distinct percentage concentration of carbohydrate. The trainer aims to mix these solutions to achieve a specific desired percentage (27%). To achieve this, correctly calculating each component's percentage contribution is key.

Mixture Problems in Mathematics

Mixture problems involve combining substances to reach a desired concentration or property. This concept is common in chemistry, pharmacology, and even everyday cooking. In these problems, you often know the concentration or percentage of the component substances and aim to find the correct amounts to mix.
For example, the exercise describes creating a 27% carbohydrate solution by mixing a 15% and a 35% solution. The mathematical challenge is to calculate the amount of each solution needed.

• Key to solving mixture problems is setting up an equation that reflects the total percentage desired.
• Here, the equation \(0.27(a + b)=0.15a + 0.35b\) represents the balance between the mixtures.

The collaboration of knowledge about percentages and algebra helps solve such problems by allowing us to focus on the relationship between different parts of a mixture and their overall effect.

Solving Algebraic Equations

Algebraic equations are statements of equality containing one or more variables. Solving them involves finding values for the variables that satisfy the equation. This process includes several steps, crucial in problems like the one being discussed.
In the given exercise, we need to solve for the variable \( b \), representing the amount of 35% solution needed:

• The initial equation derived from understanding the mixture problem is \(0.27(10 + b) = 0.15 \times 10 + 0.35b\).
• Expanding and rearranging this equation gives \(2.7 + 0.27b = 1.5 + 0.35b\).
• By isolating \( b \) on one side, we ultimately solve \(0.08b = 1.2\).
• Finally, dividing both sides by 0.08 results in \(b = 15\).

Understanding how to manipulate equations is essential in breaking down complex problems into manageable steps, leading to accurate solutions.