Answer :
- Set up the inequality $-16t^2 + 729 < 104$.
- Simplify the inequality to $t^2 > \frac{625}{16}$.
- Take the square root of both sides, resulting in $t > 6.25$ or $t < -6.25$.
- Since time must be non-negative, the solution is $t > 6.25$.
- The interval of time is $\boxed{t>6.25}$.
### Explanation
1. Problem Setup
We are given the equation $h = -16t^2 + 729$ that models Jerald's height, $h$, in feet, at time $t$ seconds after he jumped. We want to find the interval of time for which his height is less than 104 feet. This means we need to solve the inequality $-16t^2 + 729 < 104$.
2. Isolating the Quadratic Term
First, we subtract 729 from both sides of the inequality:
$$-16t^2 + 729 - 729 < 104 - 729$$
$$-16t^2 < -625$$
3. Dividing by a Negative Number
Next, we divide both sides by -16. Remember that when we divide or multiply an inequality by a negative number, we must reverse the inequality sign:
$$\frac{-16t^2}{-16} > \frac{-625}{-16}$$
$$t^2 > \frac{625}{16}$$
4. Taking the Square Root
Now, we take the square root of both sides. Remember to consider both positive and negative square roots:
$$t > \sqrt{\frac{625}{16}} \quad \text{or} \quad t < -\sqrt{\frac{625}{16}}$$
Since $\sqrt{\frac{625}{16}} = \frac{\sqrt{625}}{\sqrt{16}} = \frac{25}{4} = 6.25$, we have:
$$t > 6.25 \quad \text{or} \quad t < -6.25$$
5. Considering the Time Domain
Since $t$ represents time, it must be non-negative. Therefore, we can disregard the solution $t < -6.25$. So, we are left with $t > 6.25$.
6. Final Answer
Therefore, Jerald is less than 104 feet above the ground for $t > 6.25$ seconds.
### Examples
Understanding how an object's height changes over time is crucial in many fields. For example, engineers designing amusement park rides need to calculate the height and speed of the ride at different points to ensure safety and excitement. Similarly, in sports, understanding the trajectory of a ball or the height of a jump helps athletes optimize their performance. This problem demonstrates a basic application of quadratic equations in modeling real-world scenarios involving motion and gravity.
- Simplify the inequality to $t^2 > \frac{625}{16}$.
- Take the square root of both sides, resulting in $t > 6.25$ or $t < -6.25$.
- Since time must be non-negative, the solution is $t > 6.25$.
- The interval of time is $\boxed{t>6.25}$.
### Explanation
1. Problem Setup
We are given the equation $h = -16t^2 + 729$ that models Jerald's height, $h$, in feet, at time $t$ seconds after he jumped. We want to find the interval of time for which his height is less than 104 feet. This means we need to solve the inequality $-16t^2 + 729 < 104$.
2. Isolating the Quadratic Term
First, we subtract 729 from both sides of the inequality:
$$-16t^2 + 729 - 729 < 104 - 729$$
$$-16t^2 < -625$$
3. Dividing by a Negative Number
Next, we divide both sides by -16. Remember that when we divide or multiply an inequality by a negative number, we must reverse the inequality sign:
$$\frac{-16t^2}{-16} > \frac{-625}{-16}$$
$$t^2 > \frac{625}{16}$$
4. Taking the Square Root
Now, we take the square root of both sides. Remember to consider both positive and negative square roots:
$$t > \sqrt{\frac{625}{16}} \quad \text{or} \quad t < -\sqrt{\frac{625}{16}}$$
Since $\sqrt{\frac{625}{16}} = \frac{\sqrt{625}}{\sqrt{16}} = \frac{25}{4} = 6.25$, we have:
$$t > 6.25 \quad \text{or} \quad t < -6.25$$
5. Considering the Time Domain
Since $t$ represents time, it must be non-negative. Therefore, we can disregard the solution $t < -6.25$. So, we are left with $t > 6.25$.
6. Final Answer
Therefore, Jerald is less than 104 feet above the ground for $t > 6.25$ seconds.
### Examples
Understanding how an object's height changes over time is crucial in many fields. For example, engineers designing amusement park rides need to calculate the height and speed of the ride at different points to ensure safety and excitement. Similarly, in sports, understanding the trajectory of a ball or the height of a jump helps athletes optimize their performance. This problem demonstrates a basic application of quadratic equations in modeling real-world scenarios involving motion and gravity.
