Problem 50 - Elementary Algebra

Question

For Problems \(33-50\), set up an equation and solve the problem. (Objective 2 ) Machine \(A\) can wrap 600 pieces of candy in 5 minutes less time than it takes machine \(B\) to wrap 600 pieces of candy. If the rate of machine \(A\) is 20 candies per minute faster than machine \(B\), find the rate of each machine.

Step-by-Step Solution

Answer

Machine B: 40 candies/minute, Machine A: 60 candies/minute.

Step 1: Define Variables

Let the rate of machine B be \(r\) candies per minute. Since machine A is 20 candies per minute faster than machine B, the rate of machine A is \(r + 20\) candies per minute.

Step 2: Set Up Time Equations

The time taken by machine B to wrap 600 candies is \(\frac{600}{r}\) minutes. The time taken by machine A to wrap 600 candies is \(\frac{600}{r + 20}\) minutes. Since machine A takes 5 minutes less to wrap the candies than machine B, we have the equation \(\frac{600}{r} - \frac{600}{r + 20} = 5\).

Step 3: Solve the Equation

Start with the equation \(\frac{600}{r} - \frac{600}{r + 20} = 5\). Multiply through by \(r(r + 20)\) to clear the fractions: \[600(r + 20) - 600r = 5r(r + 20)\]Simplify this to:\[12000 = 5r^2 + 100r\] Rearrange to form a quadratic equation:\[5r^2 + 100r - 12000 = 0\]

Step 4: Solve the Quadratic Equation

Divide the entire equation by 5 to simplify:\[r^2 + 20r - 2400 = 0\]Use the quadratic formula, where \(a = 1\), \(b = 20\), \(c = -2400\):\[r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Substitute the values into the formula:\[r = \frac{-20 \pm \sqrt{20^2 - 4 \cdot 1 \cdot (-2400)}}{2 \cdot 1}\]\[r = \frac{-20 \pm \sqrt{400 + 9600}}{2}\]\[r = \frac{-20 \pm \sqrt{10000}}{2}\]\[r = \frac{-20 \pm 100}{2}\]This gives \(r = 40\) or \(r = -60\). Since a rate cannot be negative, \(r = 40\).

Step 5: Determine Rates of Machines A and B

Since \(r = 40\) candies per minute for machine B, the rate of machine A, which is 20 candies per minute faster, is \(40 + 20 = 60\) candies per minute.

Key Concepts

Quadratic Equations

Quadratic equations are equations that can be written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown.
This type of equation is fundamental in algebra and can have two solutions, one solution, or no real solutions at all.

• The solutions to a quadratic equation are often found using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
• This formula provides a way to calculate the roots (solutions) by substituting the values of \(a\), \(b\), and \(c\).
• A key part of solving a quadratic equation involves the discriminant \(b^2 - 4ac\), which determines the nature of the roots.

When solving the quadratic equation in our problem, we describe the relationship between time, rate, and distance involving machine operations.

Rate Problems

Rate problems are mathematical challenges involving rates of speed, work, or flow over time.
They often require setting up equations based on the formula \(\text{Rate} \times \text{Time} = \text{Work}\) or \(\text{Rate} \times \text{Time} = \text{Distance}\).
This basic formula helps us relate different variables such as time and quantity processed.

• In the context of machine operations, a rate problem might involve calculating how quickly a machine can accomplish a task, like wrapping candies.
• To solve these problems, we often need to determine how changes in rate affect total time or output.
• Here, since machine A wraps candies 20 candies per minute faster than machine B, this difference helped set up the original equation.

Equation Solving

Equation solving is the process of finding the value(s) of variable(s) that satisfy an equation.
The primary goal is to manipulate the equation to isolate the unknown variable.
Equations can be linear, quadratic, or even more complex.

• In this task, solving the equation involved clearing fractions by multiplying both sides by the same value, rearranging terms, and simplifying.
• Once simplified to a quadratic equation, we applied the quadratic formula to find potential solutions.
• Choosing valid solutions is crucial, given context—for example, disregarding negative rates as they don't make sense in real-world applications.

Understanding and applying operations like addition, subtraction, multiplication, and division are key to solving equations effectively.

Mathematical Modeling

Mathematical modeling is a process where a real-world situation is transformed into a mathematical format, such as equations or formulas, to analyze and solve problems.
It requires identifying critical variables and relationships within the scenario.

• First, define the variables: for instance, the rates of the machines were labeled as \(r\) and \(r + 20\) based on given differences.
• Next, write mathematical equations to express relationships. Here, the time equation \(\frac{600}{r} - \frac{600}{r + 20} = 5\) represented the difference in operational time between two machines.
• Finally, use these equations to interpret results and draw conclusions about the rates of the machines.

Mathematical modeling connects abstract mathematical concepts with tangible real-world applications, making algebra practical and applicable.