To determine how many seconds the riders fall before slowing down, we need to use the given height formula:
\[ h = -16t^2 + s \]
In this formula, \( h \) is the height after \( t \) seconds, and \( s \) is the initial height from which the riders fall.
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Identify the variables:
- The starting height \( s \) is 200 feet (the height of the tower).
- The riders fall to a height of 75 feet before the brakes are activated.
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Set up the equation: We want to find \( t \) when the height \( h \) is 75 feet. So, we can set up the equation:
\[ 75 = -16t^2 + 200 \]
- Rearranging the equation: Subtract 200 from both sides:
\[ 75 - 200 = -16t^2 \]
\[ -125 = -16t^2 \]
- Divide by -1 to simplify:
\[ 125 = 16t^2 \]
- Solve for \( t^2 \):
\[ t^2 = \frac{125}{16} \]
- Taking the square root:
\[ t = \sqrt{\frac{125}{16}} \]
Calculating that:
\[ t = \frac{\sqrt{125}}{4} = \frac{5\sqrt{5}}{4} \]
- Now, compute \( \sqrt{125} \):
Since \( \sqrt{125} = 11.1803 \) (approximately),
\[ t \approx \frac{11.1803}{4} \approx 2.795 \]
- Round to the nearest tenth:
\[ t \approx 2.8 \]
Therefore, the riders fall for approximately 2.8 seconds before slowing down.