Jerald's Bungee Jump: Calculating Time Below 104 Feet

Hey everyone! Today, we're diving into a cool math problem involving Jerald's daring bungee jump. We'll be using a quadratic equation to figure out when he's less than 104 feet above the ground. This is a great example of how math can be applied to real-life scenarios, making it more fun and relatable. Let's break it down step by step to make sure we get this right, alright?

Understanding the Problem: Jerald's Bungee Jump and Height Equation

First off, let's understand the scenario. Jerald takes a leap from a bungee tower, and we're given an equation that models his height. This equation is: $h = -16t^2 + 729$, where h represents the height in feet and t represents the time in seconds. The question asks us to determine the time interval during which Jerald is less than 104 feet above the ground. This means we need to find the range of time (t) when his height (h) is less than 104 feet. This is a classic example of using a quadratic equation to solve a problem related to motion under the influence of gravity. The negative sign in front of the $16t^2$ term indicates that the acceleration due to gravity is pulling Jerald downwards, causing his height to decrease over time after he jumps. The constant term, 729, represents the initial height of the tower, which is where Jerald starts his jump. The quadratic nature of the equation tells us that the path Jerald takes is a parabola, with the highest point being the initial height of the tower. As Jerald falls, his height decreases until the bungee cord reaches its limit, causing him to bounce back up. So, understanding the equation's components is key to solving the problem. We are looking for the time interval during which Jerald's height is less than 104 feet. This involves solving an inequality, which we'll cover in the next section, to find the specific values of t that satisfy this condition. The problem also provides answer choices in the form of inequalities, meaning we must determine which interval correctly represents the time during which Jerald is below 104 feet. By understanding the nature of the quadratic equation and what the variables represent, we can approach this problem logically and accurately.

Now that we know all that, let's figure out how to actually solve it!

Solving the Inequality: Finding the Time Interval

Okay, guys, here's where the rubber meets the road! To find the time interval when Jerald is less than 104 feet, we need to set up an inequality. We know that $h < 104$. We also know that $h = -16t^2 + 729$. So, we can substitute the expression for h into our inequality, giving us: $-16t^2 + 729 < 104$. Now, we need to solve this inequality for t. This means we will manipulate the equation to isolate t on one side. First, subtract 104 from both sides of the inequality to move everything to one side: $-16t^2 + 625 < 0$. Now, to make it easier to work with, let's isolate the t term. Subtract 729 from both sides: $-16t^2 < 104 - 729$ which simplifies to $-16t^2 < -625$. Next, divide both sides by -16. Remember, when you divide or multiply both sides of an inequality by a negative number, you must flip the inequality sign. So, we get: t^2 > rac{625}{16}. Now we take the square root of both sides to solve for t. Remember, when you take the square root, you need to consider both the positive and negative roots: t > rac{25}{4} or t < -rac{25}{4}. These values can be written as $t > 6.25$ or $t < -6.25$. However, since time cannot be negative in this context (it doesn't make sense for Jerald to have jumped before he jumped!), we only consider the positive value. Therefore, the time interval when Jerald is less than 104 feet above the ground is $t > 6.25$ seconds. This inequality indicates that Jerald will be below 104 feet after 6.25 seconds. Also it's critical to remember the properties of inequalities and how they are affected by mathematical operations. The process of isolating t and keeping track of the inequality sign's direction is essential for finding the correct solution. Thus, this process will ensure that we're on the right track and get the correct answer.

Selecting the Correct Answer: Time and Context

Alright, let's look at our answer choices. We've determined that Jerald is less than 104 feet above the ground when $t > 6.25$ seconds. Now, let's compare this with the given options.

• Option A: $t > 6.25$
• Option B: $-6.25 < t < 6.25$
• Option C: $t < 6.25$
• Option D: $0 extless t extless 6.25$

Looking at our solution and the answer choices, it's clear that option A matches our result: $t > 6.25$. This means Jerald is less than 104 feet above the ground after 6.25 seconds. The other options do not match the context of the bungee jump. Option B would mean Jerald is under 104 feet for a specific time. Option C indicates that Jerald is under 104 feet before the jump, and option D is the time when Jerald is above the ground. Therefore, A is the best answer since it accurately reflects the time frame during which Jerald's height is below 104 feet. This means that we only choose the option that aligns with the mathematical solution and the real-world situation. This ensures that the chosen answer is not just mathematically correct, but also makes practical sense in the context of the bungee jump.

Conclusion: Jerald's Journey and Mathematical Application

Awesome, guys! We've successfully figured out the time interval when Jerald is less than 104 feet above the ground. We used a quadratic equation to model his height and solved an inequality to find the answer. This problem is a great example of how math is used to describe and predict real-world scenarios. Understanding how to set up and solve such problems can be super helpful. Keep practicing, and you'll get better at these types of questions. Always remember the fundamental concepts: setting up the inequality correctly, isolating the variable, and considering the context of the problem. We've applied these mathematical concepts to find the answer in this problem. Always double-check the reasonableness of your answers within the context of the problem. It's fun to see how these concepts can be applied and make the world around us a little clearer!