Absolutely, let's work through the problem step by step.
Given:
- Mass of the object ([tex]\( m \)[/tex]): [tex]\( 0.100 \, \text{kg} \)[/tex]
- Potential Energy ([tex]\( PE \)[/tex]): [tex]\( 179.99 \, \text{Joules} \)[/tex]
- Acceleration due to gravity ([tex]\( g \)[/tex]): [tex]\( 9.8 \, \text{m/s}^2 \)[/tex]
We want to determine the height ([tex]\( h \)[/tex]) from which the object was dropped. We can use the relationship between potential energy, mass, height, and gravity:
[tex]\[ PE = m \cdot g \cdot h \][/tex]
We solve for [tex]\( h \)[/tex]:
[tex]\[ h = \frac{PE}{m \cdot g} \][/tex]
Substitute the given values into the formula:
[tex]\[ h = \frac{179.99 \, \text{J}}{0.100 \, \text{kg} \cdot 9.8 \, \text{m/s}^2} \][/tex]
[tex]\[ h = \frac{179.99}{0.100 \cdot 9.8} \][/tex]
[tex]\[ h = \frac{179.99}{0.98} \][/tex]
[tex]\[ h = 183.66326530612244 \, \text{meters} \][/tex]
Now, compare this result to the given choices:
A. [tex]\( 9.2 \times 10^1 \)[/tex] meters [tex]\( = 92 \)[/tex] meters
B. [tex]\( 1.8 \times 10^2 \)[/tex] meters [tex]\( = 180 \)[/tex] meters
C. [tex]\( 9.8 \times 10^2 \)[/tex] meters [tex]\( = 980 \)[/tex] meters
D. [tex]\( 1.0 \times 10^2 \)[/tex] meters [tex]\( = 100 \)[/tex] meters
From these options, the closest one to our calculated height of approximately [tex]\( 183.663 \)[/tex] meters is B. [tex]\( 1.8 \times 10^2 \)[/tex] meters.
Therefore, the correct answer is:
B. [tex]\( 1.8 \times 10^2 \)[/tex] meters
