Applications of Non-Linear Equations: Learn It 3

Real-world Applications

Understanding and applying non-linear equations is crucial in many fields, including physics, engineering, economics, and natural sciences. They allow us to model and analyze complex phenomena that don’t follow simple linear patterns.

The period of a pendulum, [latex]T[/latex] is modeled by
[latex]T = 2 \pi \sqrt{\dfrac{L}{g}}[/latex]
where the length of the pendulum is [latex]L[/latex] and the acceleration due to gravity is [latex]g[/latex].If the acceleration due to gravity is [latex]9.8 \frac{\text{m}}{\text{s}^2}[/latex] and the period equals [latex]1 s[/latex], find the length to the nearest cm.

Joe and John are planning to paint a house together.

  • John thinks that if he worked alone, it would take him 3 times as long as it would take Joe to paint the entire house.
  • Working together, they can complete the job in 24 hours.

How long would it take each of them, working alone, to complete the job?

To find out how long it would take Joe and John to paint the house individually, we start by setting up an equation based on their working rates and then solve for the individual times.

  • Joe’s rate of painting is [latex]\frac{1}{t}[/latex] of the house per hour.
  • John’s rate of painting is [latex]\frac{1}{3t}[/latex] of the house per hour.

Together, they finish the job in [latex]24[/latex] hours. Thus, their combined rate is: [latex]\frac{1}{24}[/latex] of the house per hour.

Their rates add up to the total rate:

[latex]\frac{1}{t} + \frac{1}{3t} = \frac{1}{24}[/latex]

We can use this to find the time for Joe.

[latex]\begin{align*} \text{Combine individual rates} & : & \frac{1}{t} + \frac{1}{3t} &= \frac{1}{24} & \\ \text{Simplify the equation} & : & \frac{4}{3t} &= \frac{1}{24} & \\ \text{Clear the fraction by cross-multiplying} & : & 4 \times 24 &= 3t & \\ \text{Calculate the product} & : & 96 &= 3t & \\ \text{Solve for } t & : & t &= \frac{96}{3} & \\ \text{Simplify to find the time for Joe alone} & : & t &= 32 \text{ hours} & \end{align*}[/latex]

Now we can determine John’s time.

[latex]3t = 96 \text{ hours}[/latex]

Thus, Joe would need [latex]32[/latex] hours and John would need [latex]96[/latex] hours to paint the house alone.