To approximate the height of the Sears Tower using the given formula, follow these steps:
1. Identify the known values:
- The time, [tex]\( t \)[/tex], it takes for the object to reach the ground (height [tex]\( h = 0 \)[/tex] feet) is given as 9.7 seconds.
- The acceleration due to gravity in the formula is 16 feet per second squared.
2. Recall the formula:
[tex]\[
t = \sqrt{\frac{h_0 - h}{16}}
\][/tex]
where [tex]\( h_0 \)[/tex] is the initial height and [tex]\( h \)[/tex] is the final height (which is 0 in this case).
3. Plug in the known values:
Since the object is dropped, the final height [tex]\( h \)[/tex] is 0 feet. Thus, the formula simplifies to:
[tex]\[
t = \sqrt{\frac{h_0}{16}}
\][/tex]
Substitute [tex]\( t = 9.7 \)[/tex] seconds into the equation:
[tex]\[
9.7 = \sqrt{\frac{h_0}{16}}
\][/tex]
4. Solve for [tex]\( h_0 \)[/tex]:
First, square both sides of the equation to eliminate the square root:
[tex]\[
(9.7)^2 = \frac{h_0}{16}
\][/tex]
Calculate [tex]\( (9.7)^2 \)[/tex]:
[tex]\[
9.7^2 = 94.09
\][/tex]
So the equation now is:
[tex]\[
94.09 = \frac{h_0}{16}
\][/tex]
5. Isolate [tex]\( h_0 \)[/tex]:
Multiply both sides by 16 to solve for [tex]\( h_0 \)[/tex]:
[tex]\[
h_0 = 94.09 \times 16
\][/tex]
Calculate the product:
[tex]\[
h_0 = 1505.44
\][/tex]
6. Approximate to the nearest foot:
The computed height of the Sears Tower is approximately 1505.44 feet. Rounding to the nearest foot:
[tex]\[
h_0 \approx 1505 \text{ feet}
\][/tex]
Thus, the height of the Sears Tower is approximately 1505 feet.
