Question: 17. According to legend, in 1589, the Italian scientist Galileo Galilei dropped rocks of different weights from the top of the Leaning Tower of Pisa to prove his conjecture that the rocks would hit the ground at the same time.
A. The original height of the tower was about 196 feet. Write a function h that gives the height (in feet) of a rock dropped from the top of the original tower after t seconds.
a. Interpret each term.
b. How long does the rock take to hit the ground?
B. Find and interpret h(1.25) - h(2.5).
1. **State the problem:**
We want to model the height $h(t)$ of a rock dropped from the Leaning Tower of Pisa, originally 196 feet tall, after $t$ seconds.
2. **Write the function:**
The height of a freely falling object under gravity (ignoring air resistance) is given by:
$$h(t) = h_0 - \frac{1}{2}gt^2$$
where $h_0$ is the initial height and $g$ is the acceleration due to gravity.
3. **Apply values:**
Given $h_0 = 196$ feet and $g = 32$ ft/s$^2$ (standard gravity), the function is:
$$h(t) = 196 - 16t^2$$
4. **Interpret each term:**
- $196$ is the initial height of the tower in feet.
- $16t^2$ represents the distance fallen after $t$ seconds (half of $g$ times $t^2$).
5. **Find time to hit the ground:**
Set $h(t) = 0$ to find when the rock hits the ground:
$$0 = 196 - 16t^2$$
$$16t^2 = 196$$
$$t^2 = \frac{196}{16}$$
$$t^2 = 12.25$$
$$t = \sqrt{12.25} = 3.5$$
So, the rock takes $3.5$ seconds to hit the ground.
6. **Calculate $h(1.25)$ and $h(2.5)$:**
$$h(1.25) = 196 - 16(1.25)^2 = 196 - 16(1.5625) = 196 - 25 = 171$$
$$h(2.5) = 196 - 16(2.5)^2 = 196 - 16(6.25) = 196 - 100 = 96$$
7. **Find and interpret $h(1.25) - h(2.5)$:**
$$h(1.25) - h(2.5) = 171 - 96 = 75$$
This means the rock fell $75$ feet between $1.25$ seconds and $2.5$ seconds after being dropped.
**Final answers:**
- Function: $$h(t) = 196 - 16t^2$$
- Time to hit ground: $$3.5$$ seconds
- Height difference between $1.25$ and $2.5$ seconds: $$75$$ feet