The distance, d, that an object falls is directly proportional to the square of the time, t, it has been in free fall. An object that has been in free fall for 5 seconds has fallen 400 feet. Determine the distance the object has fallen if it has been falling for 3 seconds.
We can set up a proportion based on the given information:
\(d \propto t^2\)
We know that when \(t = 5\), \(d = 400\).
So, we have:
\(\frac{d}{t^2} = \frac{400}{5^2}\)
Simplifying the right side:
\(\frac{d}{t^2} = \frac{400}{25}\)
\(\frac{d}{t^2} = 16\)
Now we can find the value of \(d\) when \(t = 3\):
\(\frac{d}{3^2} = 16\)
\(\frac{d}{9} = 16\)
Multiplying both sides by 9:
\(d = 16(9)\)
\(d = 144\)
Therefore, if the object has been falling for 3 seconds, it has fallen 144 feet.