Problem 59 - Prealgebra

Question

The time \(t\) in seconds it takes an object to fall \(d\) feet can be estimated by using \(d=0.5(32) t^{2} .\) If a ball is dropped from the top of a 120 -foot building, how long does it take to hit the ground? A \(1.9 \mathrm{s}\) C \(3.8 \mathrm{s}\) B \(2.7 \mathrm{s}\) D \(7.5 \mathrm{s}\)

Step-by-Step Solution

Answer

The ball takes approximately 2.7 seconds to hit the ground, matching option B.

Step 1: Identify the Equation

The problem gives us the equation for the distance fallen as a function of time: \[d = 0.5 \times 32 \times t^2\]. This equation can be simplified to \(d = 16t^2\). We are asked to solve for \(t\), given \(d = 120\) feet.

Step 2: Substitute the Known Values

Substitute the given distance into the equation. Replace \(d\) with 120: \[120 = 16t^2\]. This equation will allow us to solve for \(t\).

Step 3: Solve for Time (t)

To isolate \(t^2\), divide both sides by 16: \[t^2 = \frac{120}{16}\]. Simplify this to: \[t^2 = 7.5\].

Step 4: Compute the Square Root

Take the square root of both sides to solve for \(t\): \[t = \sqrt{7.5}\]. Calculate this to get \(t \approx 2.74\).

Step 5: Round to Match Options

The calculated time is approximately 2.74 seconds. Round this to the nearest tenth to get \(t \approx 2.7\) seconds, which corresponds to one of the answer choices.

Key Concepts

Understanding Quadratic Equations

Quadratic equations may sound complex, but they're simply equations where the highest power of an unknown variable is squared. In general, a quadratic equation is written in the form \(ax^2 + bx + c = 0\), where \(x\) is the unknown variable. Such equations are essential in various fields because they can describe a wide range of natural phenomena, from the trajectory of a thrown object to the shapes of certain graphs.
To solve a quadratic equation, you can utilize several methods:

• Factoring: This involves expressing the quadratic as a product of two binomials.
• Completing the square: This method transforms the equation into a perfect square trinomial.
• Quadratic formula: A surefire way to find solutions is to use \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which can solve any quadratic equation.

In the context of our exercise, the equation \(d = 16t^2\) is quadratic because of the \(t^2\) term. When you rearrange this into \(16t^2 = d\) and solve for \(t\), you're applying the principles of solving quadratics.

Tackling Problem Solving in Mathematics

Problem solving is the heart of mathematics and is a skill that helps in a variety of real-world scenarios. It's about logical reasoning, pattern recognition, and a systematic approach to finding solutions. Tackling a problem typically involves several steps:

• Understanding the problem: Start by identifying what you're being asked to find and what information is provided.
• Setting up equations: Convert the problem statement into a mathematical equation, as we did by identifying \(d = 16t^2\) and substituting the given \(d = 120\).
• Solving the equations: This involves manipulation to isolate the variable and find its value, such as dividing both sides by 16 to find \(t^2\).
• Verifying the solution: Check that the solution makes sense within the context of the problem. In our example, compute \(t\) to ensure it's a feasible time duration.

Improving problem-solving skills involves practice and developing an intuition for when and how to apply different mathematical techniques.

The Role of Physics in Mathematics

Physics and mathematics go hand in hand, often intertwining to explain how objects move and interact with natural forces. In physics, equations like the one in our exercise represent real-world phenomena—specifically, they model how an object falls due to gravity.
The equation \(d = 16t^2\) stems from the physics formula for distance fallen under gravity, \(d = \frac{1}{2}gt^2\), where \(g\) is the acceleration due to gravity (32 feet per second squared on Earth). This reflects the effect of gravity pulling the object downward, increasing its speed over time.
Understanding the interplay between physics and math is crucial for real-world applications:

• Predicting how long it will take an object to fall to the ground, as in our problem.
• Designing structures that can withstand gravitational forces.
• Exploring concepts of motion, energy, and force in greater depth.

By grasping the basic physics in mathematics, you can appreciate how such equations not only solve textbook problems but also explain the world around us.