Problem 112 - Elementary and Intermediate Algebra

Question

The distance in feet that an object will fall in \(t\) seconds is given by the expression \(16 t^{2} .\) Find the distance that riders on "Drop Zone" will fall during the times listed in the table. $$ \begin{array}{|c|c|} \hline \text { Time } & {\text { Distance }} \\ \hline \text { (seconds) } & {\text { (feet) }} \\ \hline 1 & {} \\ \hline 2 & {} \\ \hline 3 & {} \\ \hline 4 & {} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

At 1s: 16ft, 2s: 64ft, 3s: 144ft, 4s: 256ft.

Step 1: Understanding the Formula

The formula given for the distance fallen by an object in time \(t\) seconds is \(16t^2\). This means the distance is directly proportional to the square of the time.

Step 2: Calculating Distance for 1 Second

For \(t = 1\) second, substitute into the formula: \(16t^2 = 16(1)^2 = 16 \text{ feet}\).

Step 3: Calculating Distance for 2 Seconds

For \(t = 2\) seconds, substitute into the formula: \(16t^2 = 16(2)^2 = 16 \times 4 = 64 \text{ feet}\).

Step 4: Calculating Distance for 3 Seconds

For \(t = 3\) seconds, substitute into the formula: \(16t^2 = 16(3)^2 = 16 \times 9 = 144 \text{ feet}\).

Step 5: Calculating Distance for 4 Seconds

For \(t = 4\) seconds, substitute into the formula: \(16t^2 = 16(4)^2 = 16 \times 16 = 256 \text{ feet}\).

Key Concepts

Distance Formula

Calculating the distance an object falls over time involves an important mathematical tool: the distance formula. For this specific scenario, the formula given is \( 16t^2 \). This tells us that the distance an object falls is directly influenced by the square of the time in seconds. It means that as time increases, the distance grows rapidly. This formula is derived from physics, where the constant 16 represents half of the acceleration due to gravity in feet per second squared, a standard approximation used for objects in free fall.

• This formula simplifies complex physics into a manageable calculation.
• Provides a clear example of quadratic relationships.

The formula is intuitive once you understand the role each part plays—the constant 16 helps convert the square of time into a real-world measurement, distance. As such, this formula helps us clearly determine how far an object will fall over a given time.

Proportional Relationships

In this context, a proportional relationship exists between the time it takes for an object to fall and the square of that time, as indicated by the expression \( 16t^2 \). This means if you double the time, you multiply the squared time by four, causing the distance to increase by a factor of four — a characteristic of squaring a variable in mathematical equations.
This demonstrates a key feature of quadratic functions:

• Quadratic relationships are not linear, they curve upwards.
• They produce a parabolic graph, showing rapid increases in distance with each increment of time.

Such proportional relationships help us model and predict real-world scenarios where changes do not happen at a constant rate, but instead accelerate as time progresses. Understanding this concept is crucial in interpreting situations where variables are interdependent in complex ways.

Time and Distance Calculations

Time and distance calculations in this exercise involve applying a quadratic equation to solve practical problems. The formula \( 16t^2 \) guides you on how to substitute values properly.
For example:

• At \( t = 1 \) second, the distance is \( 16(1)^2 = 16 \) feet, meaning the drop is only at its start.
• As \( t = 2 \) seconds, \( 16(2)^2 = 64 \) feet is calculated, quadruple the distance at \( t = 1 \), neatly showcasing the squared term's effect.
• At \( t = 3 \), the object falls \( 16(3)^2 = 144 \) feet - a substantial increase.
• Finally, measuring at \( t = 4 \), the distance becomes \( 16(4)^2 = 256 \) feet - illustrating massive growth in fall distance over just a short span of time.

This demonstrates how each second adds more distance than the last—an essential insight that shows how quadratic functions model acceleration and growth in real-life scenarios. Learning such calculations can assist you in both academic scenarios and understanding real-world phenomena involving acceleration.