Let's go through each of the questions step-by-step.

Question 1

The problem gives a height function for a ride:

$h = -16t^2 + 64t + 60$

where $h$ is the height in feet, and $t$ is the time in seconds. The task is to determine how many seconds it takes for the ride to reach ground level, which means finding $t$ when $h = 0$.

1. Set up the equation by substituting $h = 0$: $0 = -16t^2 + 64t + 60$

2. Rearrange the equation to standard quadratic form: $-16t^2 + 64t + 60 = 0$

3. Use the quadratic formula $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = -16$, $b = 64$, and $c = 60$.

4. Substitute these values into the formula to solve for $t$. Your work on the page shows calculation steps using this approach.

After simplifying, you should get two possible values for $t$. Only the positive value will be meaningful in this context (as time cannot be negative).

Question 2

The area of a rectangle is 117 square meters. The rectangle's length is 4 meters more than its width.

1. Let the width be $w$ and the length be $w + 4$.

2. Set up the equation for the area: $w(w + 4) = 117$

3. Expanding and rearranging gives: $w^2 + 4w - 117 = 0$

4. Use the quadratic formula to solve for $w$: $w = \frac{-4 \pm \sqrt{4^2 + 4 \cdot 117}}{2}$

5. After finding $w$, calculate $w + 4$ to get the length.

Question 3

The product of two consecutive negative integers is 1122. Let the integers be $x$ and $x+1$.

1. Set up the equation: $x(x + 1) = 1122$

2. This simplifies to: $x^2 + x - 1122 = 0$

3. Use the quadratic formula to solve for $x$: $x = \frac{-1 \pm \sqrt{1 + 4 \cdot 1122}}{2}$

4. Determine the values of $x$ and $x + 1$.

Would you like me to go through the calculations in detail for each question?

Follow-Up Questions

1. Would you like further clarification on how to use the quadratic formula in different contexts?
2. Do you want to know how to check your answers for these types of problems?
3. Are there any specific steps in solving quadratic equations you find challenging?
4. Would you like more examples of word problems involving quadratic equations?
5. Are you interested in a general method for solving quadratic equations without using the formula?

Tip: Always check your discriminant (the value inside the square root of the quadratic formula) to see if your equation has real solutions before solving.