Problem 360 - Elementary Algebra

Question

A watermelon is dropped from the tenth story of a building. Solve the equation \(-16 t^{2}+144=0\) for \(t\) to find the number of seconds it takes the watermelon to reach the ground.

Step-by-Step Solution

Answer

It takes 3 seconds for the watermelon to reach the ground.

Step 1: Identify the given equation

The given equation to solve is \(-16 t^{2} + 144 = 0\).

Step 2: Move the constant term to the other side

Subtract 144 from both sides to isolate the term with the variable: \(-16 t^{2} + 144 - 144 = 0 - 144\) which simplifies to \(-16 t^{2} = -144\).

Step 3: Divide by the coefficient of the squared term

Divide both sides by -16 to solve for \(t^{2}\): \(\frac{-144}{-16} = t^{2}\). This simplifies to \(t^{2} = 9\).

Step 4: Solve for the variable

Take the square root of both sides to solve for \(t\): \(\begin{array}{c}\ t = \text{±} \sqrt{9 \) \end{array}\ which gives two solutions, \(t = 3\) and \(t = -3\).

Step 5: Consider the physical context

Since time cannot be negative, the only feasible solution is \(t = 3\) seconds.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest power of the variable is 2. They follow the general form:
\(ax^2 + bx + c = 0\).
Here, \(a\), \(b\), and \(c\) are constants (with \(a ≠ 0\)).
A quadratic equation can have potential outcomes: two real roots, one real root, or no real roots.
It is important to understand how to manipulate these equations to find their solutions.

There are different methods to solve quadratic equations:

• **Factoring**: This method involves expressing the quadratic equation as a product of its factors.
• **Graphing**: Here, you plot the quadratic equation and identify where it crosses the x-axis.
• **Using the Quadratic Formula**: This formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), can solve any quadratic equation by substituting the values of \(a\), \(b\), and \(c\).
• **Completing the Square**: This allows rewriting the quadratic equation in such a way that makes it easier to solve.

Understanding these techniques will give you the tools you need to solve quadratic equations, such as finding the time it takes for a projectile to hit the ground, as shown in the problem.

Solving Equations

Solving equations involves finding the value(s) of the variable that make the equation true. **Linear equations** are straightforward, involving variables to the first power, but quadratic equations involve squared terms, presenting more complexity.

Let's break down the steps for the given quadratic equation: \( -16t^2 + 144 = 0 \) to find the time \(t\):

• **Step 1**: Identify the equation. Here, we have \( -16t^2 + 144 = 0 \).
• **Step 2**: Move the constant term to isolate the variable term. We subtract \(144\) from both sides: \( -16t^2 = -144 \).
• **Step 3**: Divide by the coefficient of \(t^2\). Here, \( -16\) is the coefficient and dividing gives: \( t^2 = 9 \).
• **Step 4**: Solve for the variable \(t\). We take the square root of both sides, remembering both positive and negative roots, resulting in \( t = \pm 3 \).
• **Step 5**: Consider physical context. Since time cannot be negative, \( t = 3 \) seconds is the feasible solution.

Understanding these steps aids in systematically approaching any quadratic equation, ensuring all potential solutions are considered.

Physics Applications

Physics often uses equations, including quadratic equations, to describe natural phenomena. In the given problem, we applied a quadratic equation to understand the motion of a falling object. This is part of **kinematics**, the branch of mechanics that describes the motion of objects.

**Key Concepts in Physics Related to Quadratic Equations**:

• **Gravity**: When objects fall under the influence of gravity, their motion can be described by quadratic equations. Here, \( -16t^2 + 144 = 0 \) represents the vertical motion of a watermelon.
• **Displacement**: It is the change in position. In many physics problems, displacement can be related to time using quadratic equations, especially in free-fall problems.
• **Initial Velocity and Acceleration**: These can be factors in the equations. The watermelon had \(0 \) initial velocity, and \( -16\) represents half of the acceleration due to gravity (\( -32 \) feet per second\textsuperscript{2} in this example).

By solving the quadratic equation, we determined that the watermelon takes \( 3 \) seconds to fall.
Understanding how to set up and solve these equations is crucial in physics, as they help explain and predict real-world behaviors.