Jerald Jumped From A Bungee Tower. If The Equation That Models His Height, In Feet, Is $h = -16t^2 + 729$, Where $t$ Is The Time In Seconds, For Which Interval Of Time Is He Less Than 104 Feet Above The Ground?A. $t \

Introduction

Bungee jumping is an exhilarating adventure sport that involves jumping from a great height while attached to a bungee cord. The thrill of the jump is matched only by the fear of the unknown, and the physics involved in the jump can be quite complex. In this article, we will explore the equation that models Jerald's height, in feet, as he jumps from a bungee tower. We will use this equation to determine the interval of time for which Jerald is less than 104 feet above the ground.

The Equation

The equation that models Jerald's height, in feet, is given by:

$$h = -16t^2 + 729$$

where $t$ is the time in seconds. This equation is a quadratic equation, which means that it can be written in the form:

$$h = at^2 + bt + c$$

where $a$, $b$, and $c$ are constants. In this case, $a = -16$, $b = 0$, and $c = 729$.

Understanding the Equation

To understand the equation, let's break it down into its components. The term $-16t^2$ represents the acceleration due to gravity, which is acting in the opposite direction to the jump. The term $729$ represents the initial height of the tower, which is 729 feet.

Solving for Time

To find the interval of time for which Jerald is less than 104 feet above the ground, we need to solve the inequality:

$$-16t^2 + 729 < 104$$

Subtracting 729 from both sides gives us:

$$-16t^2 < -625$$

Dividing both sides by -16 gives us:

$$t^2 > 39.0625$$

Taking the square root of both sides gives us:

$$t > \sqrt{39.0625}$$

or

$$t < -\sqrt{39.0625}$$

Since time cannot be negative, we can ignore the negative solution.

Calculating the Time Interval

To calculate the time interval, we need to find the value of $t$ that satisfies the inequality:

$$t > \sqrt{39.0625}$$

Using a calculator, we find that:

$$\sqrt{39.0625} \approx 6.25$$

Therefore, Jerald is less than 104 feet above the ground for the interval:

$$t > 6.25$$

Conclusion

In conclusion, Jerald is less than 104 feet above the ground for the interval $t > 6.25$ seconds. This means that he will be below 104 feet for at least 6.25 seconds after he jumps from the tower.

Discussion

The equation that models Jerald's height is a quadratic equation, which means that it can be written in the form:

$$h = at^2 + bt + c$$

where $a$, $b$, and $c$ are constants. In this case, $a = -16$, $b = 0$, and $c = 729$.

The equation can be solved using the quadratic formula:

$$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

However, in this case, we can solve the inequality using algebraic manipulation.

References

• [1] "Bungee Jumping: The Physics Behind the Thrill". Scientific American.
• [2] "Quadratic Equations: A Guide to Solving and Graphing". Math Is Fun.

Glossary

• Bungee jumping: an adventure sport that involves jumping from a great height while attached to a bungee cord.
• Quadratic equation: an equation of the form $h = at^2 + bt + c$, where $a$, $b$, and $c$ are constants.
• Acceleration due to gravity: the acceleration due to gravity, which is acting in the opposite direction to the jump.