To find the kinetic energy (KE) of the roller-coaster car at the bottom of the hill, we can use the kinetic energy formula:
\[ KE = \frac{1}{2} mv^2 \]
where:
- \( m \) is the mass of the car and its passengers,
- \( v \) is the velocity of the car at the bottom of the hill.
Given:
- \( m = 1088 , \text{kg} \)
- \( v = 33 , \text{m/s} \)
Now, substitute the values into the equation:
\[ KE = \frac{1}{2} \times 1088 , \text{kg} \times (33 , \text{m/s})^2 \]
Calculating \( (33 , \text{m/s})^2 \):
\[ (33 , \text{m/s})^2 = 1089 , \text{m}^2/\text{s}^2 \]
Now plug that back into the kinetic energy equation:
\[ KE = \frac{1}{2} \times 1088 , \text{kg} \times 1089 , \text{m}^2/\text{s}^2 \]
Calculating this step-by-step:
- Multiply \( 1088 \) kg by \( 1089 \) m²/s²:
\[ 1088 \times 1089 = 1,180,992 , \text{kg} \cdot \text{m}^2/\text{s}^2 \]
- Then multiply by \( \frac{1}{2} \):
\[ KE = \frac{1}{2} \times 1,180,992 , \text{kg} \cdot \text{m}^2/\text{s}^2 = 590,496 , \text{J} \]
Therefore, the correct answer is:
\[ KE = 590,496 , \text{J} \]
Among the options provided, none of them match this exact value. Please double-check the calculations or the available answer choices. However, based on the calculation above, the kinetic energy at the bottom is \( 590,496 , \text{J} \).