It appears that you're describing a situation in which an object is thrown from a height and you want to track its height over time. To analyze this scenario, we can use the following general quadratic equation for projectile motion:
\[ y(t) = -at^2 + bt + h_0 \]
where:
- \( y(t) \) is the height at time \( t \).
- \( h_0 \) is the initial height (in this case, 48 feet).
- \( a \) and \( b \) are constants related to the motion (acceleration due to gravity, initial velocity, etc.).
Given that \( y(2.5) = 140 \), we can set up the equation:
- From the context, we know the initial height \( h_0 = 48 \) feet.
- The second piece of data gives us a height of 140 feet at \( t = 2.5 \) seconds, which leads to:
\[ y(2.5) = -a(2.5)^2 + b(2.5) + 48 = 140 \]
Simplifying this, we get:
\[ -a(6.25) + 2.5b + 48 = 140 \]
Subtracting 48 from both sides:
\[ -a(6.25) + 2.5b = 92 \]
So, we now have one equation. However, we need either more information or further assumptions (like the value of acceleration due to gravity, which is approximately \( 32 , \text{ft/s}^2 \) downward).
To fully solve this, we would typically have another point of reference or condition (like the initial upward velocity when Paul was thrown).
Could you provide additional information or clarify any part of the scenario?