Problem 62 - Algebra A Combined Function

Question

Together, Noodles and Freckles eat a 50 -pound bag of dog food in 30 days. Noodles by herself eats a 50-pound bag in 2 weeks less time than Freckles does by himself. How many days to the nearest whole day would a 50 -pound bag of dog food last Freckles?

Step-by-Step Solution

Answer

Freckles would take about 68 days.

Step 1: Define Variables

Let's denote the time it takes for Freckles to eat a 50-pound bag of dog food by himself as \( f \) days. Therefore, Noodles eats a 50-pound bag of dog food in \( f - 14 \) days, since she takes 2 weeks (or 14 days) less than Freckles.

Step 2: Determine Noodles and Freckles' Rate

The eating rate for Freckles is \( \frac{50}{f} \) pounds per day, while Noodles eats \( \frac{50}{f - 14} \) pounds per day.

Step 3: Combined Eating Rate Equation

Together, Noodles and Freckles can eat a 50-pound bag in 30 days. Thus, the combined rate is \( \frac{50}{f} + \frac{50}{f - 14} = \frac{50}{30} \).

Step 4: Solve the Equation

Set up the equation: \( \frac{50}{f} + \frac{50}{f - 14} = \frac{50}{30} \). Simplify and solve for \( f \):1. Multiply the entire equation by the LCD \( 30f(f-14) \): \[ 1500(f-14) + 1500f = 50f(f-14) \]2. Simplify and distribute: \[ 1500f - 21000 + 1500f = 50f^2 - 700f \]3. Combine like terms: \[ 3000f - 21000 = 50f^2 - 700f \]4. Rearrange the equation to form a quadratic: \[ 50f^2 - 3700f + 21000 = 0 \]5. Divide by 50 to simplify: \[ f^2 - 74f + 420 = 0 \]6. Solve using the quadratic formula: \( f = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -74, c = 420 \). \[ f = \frac{74 \pm \sqrt{74^2 - 4 \times 1 \times 420}}{2} \] \[ f = \frac{74 \pm \sqrt{5476 - 1680}}{2} \] \[ f = \frac{74 \pm \sqrt{3796}}{2} \] \[ f = \frac{74 \pm 61.6}{2} \] Give the two solutions and choose the positive one relevant to our context: \[ f = \frac{74 + 61.6}{2} = 67.8 \approx 68 \]

Step 5: Conclusion

Based on calculations, Freckles would take approximately 68 days to eat a 50-pound bag of dog food by himself.

Key Concepts

Rate Problems

Rate problems are all about understanding how fast or slow something occurs over time. In problems involving rates, such as how quickly Noodles and Freckles can eat a bag of dog food, it is vital to clearly define what the rate is. A rate typically describes how much of something happens in a certain time period. In this case, it's about consumption per day.

• Freckles' rate is determined by how many pounds he eats in one day, denoted as \( \frac{50}{f} \) pounds per day.
• Noodles' rate is similarly defined as \( \frac{50}{f - 14} \) pounds per day, since she is faster.

Understanding these rates allows us to combine their eating speeds and solve how they manage food together. Knowing how to work with rates also opens up your ability to solve real-world problems over time, such as travel distances or speeds.

Quadratic Equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). They are fundamental in algebra because they often describe curves, like the path of a thrown ball. In our exercise, the equation formed after organizing the rates involves solving a quadratic:

• The quadratic we derived is \( f^2 - 74f + 420 = 0 \).
• To solve it, we apply the quadratic formula: \( f = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

The quadratic formula helps find the values of \( f \) that satisfy the equation. Quadratic equations usually have two solutions, since the square term \( f^2 \) creates two intersection points. In this situation, we select the positive solution, as time cannot be negative.

System of Equations

A system of equations involves multiple equations that need to be solved together. This occurs in many real-world scenarios, like optimizing processes or distributing resources. In the Noodles and Freckles problem, their rates form a system, because we need to utilize both rates simultaneously to find the solution:

• We use the equations \( \frac{50}{f} + \frac{50}{f - 14} = \frac{50}{30} \) to solve the problem.
• This system involves finding a common solution where the two individual rates add up to the combined rate to eat the entire bag in 30 days.

The challenge is to manipulate these equations through algebra to isolate and solve for \( f \). Systems like this often require additional steps such as substitution or elimination to find compatible solutions that satisfy all conditions.