To determine which object is on the tallest hill, we need to calculate the height at which each object is located. This can be done using the formula for gravitational potential energy:
[tex]\[ PE = m \cdot g \cdot h \][/tex]
Where:
- [tex]\(PE\)[/tex] is the potential energy,
- [tex]\(m\)[/tex] is the mass,
- [tex]\(g\)[/tex] is the acceleration due to gravity ([tex]\(9.8 \, \text{m/s}^2\)[/tex]),
- [tex]\(h\)[/tex] is the height.
We can rearrange this formula to solve for height:
[tex]\[ h = \frac{PE}{m \cdot g} \][/tex]
Let's calculate the heights for each object using the provided data:
1. Object W:
[tex]\[
PE_W = 980 \, \text{J}, \quad m_W = 50 \, \text{kg}
\][/tex]
[tex]\[
h_W = \frac{980}{50 \cdot 9.8} \approx 2.00 \, \text{m}
\][/tex]
2. Object X:
[tex]\[
PE_X = 1029 \, \text{J}, \quad m_X = 35 \, \text{kg}
\][/tex]
[tex]\[
h_X = \frac{1029}{35 \cdot 9.8} \approx 3.00 \, \text{m}
\][/tex]
3. Object Y:
[tex]\[
PE_Y = 1519 \, \text{J}, \quad m_Y = 62 \, \text{kg}
\][/tex]
[tex]\[
h_Y = \frac{1519}{62 \cdot 9.8} \approx 2.50 \, \text{m}
\][/tex]
4. Object Z:
[tex]\[
PE_Z = 1176 \, \text{J}, \quad m_Z = 24 \, \text{kg}
\][/tex]
[tex]\[
h_Z = \frac{1176}{24 \cdot 9.8} \approx 5.00 \, \text{m}
\][/tex]
From these calculations, we can see that the heights for the objects are as follows:
- h_W = 2.00 meters
- h_X = 3.00 meters
- h_Y = 2.50 meters
- h_Z = 5.00 meters
The object on the tallest hill is [tex]\(Z\)[/tex], which is at a height of 5.00 meters. Therefore, the object on the tallest hill is [tex]\( \boxed{Z} \)[/tex].
