I can assist with analyzing the math problem in your image. Here is a breakdown of the questions from the image you provided:

Problem Statement: An elevator is 180 feet above the first floor. Each second it descends 12 feet.

Questions:

1. Part a: What integer is the change in the height of the elevator each second?
2. Part b: Copy and complete the table.
   • There is a table with columns "Time" and "Height" and rows corresponding to 3, 6, and 9 seconds.
3. Part c: Estimate how many seconds it takes the elevator to get to the first floor. Explain your reasoning.
4. Part d: From the first floor, it takes 4 seconds to reach the basement floor. What is the height of the basement floor with respect to the first floor?

Let's go through them step-by-step:

Part a:

• The elevator descends 12 feet each second, so the change in the elevator's height each second is -12 feet (since it's descending).

Part b:

To complete the table, we use the fact that the elevator descends at a rate of 12 feet per second.

• After 3 seconds:
  Height change = $3 \times 12 = 36$ feet
  New height = $180 - 36 = 144$ feet

• After 6 seconds:
  Height change = $6 \times 12 = 72$ feet
  New height = $180 - 72 = 108$ feet

• After 9 seconds:
  Height change = $9 \times 12 = 108$ feet
  New height = $180 - 108 = 72$ feet

So, the table will look like:

Part c:

• To find how many seconds it takes to reach the first floor, we need to calculate the total time required to descend from 180 feet to 0 feet. Since the elevator descends 12 feet each second:

$\text{Time} = \frac{\text{Height}}{\text{Rate of descent}} = \frac{180 \text{ feet}}{12 \text{ feet/second}} = 15 \text{ seconds}$

So, it will take 15 seconds to reach the first floor.

Part d:

• The elevator takes 4 seconds to reach the basement floor from the first floor. If the elevator descends 12 feet per second, the total descent during these 4 seconds is:

$4 \times 12 = 48 \text{ feet}$

Therefore, the height of the basement floor is 48 feet below the first floor, or -48 feet with respect to the first floor.

Would you like more details on any specific part of this explanation? Here are some follow-up questions:

1. How would the table change if the elevator descends at a different rate?
2. What would happen if the elevator descends faster after 10 seconds?
3. Can you calculate the distance covered in fractions of seconds?
4. What is the mathematical formula for calculating the time to reach any specific height?
5. How would you calculate the height after 12.5 seconds?

Tip: Always track units (feet, seconds) carefully in word problems to ensure accurate results.