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.