Question

the distance s that an object falls is directly proportional to the square of the time t of the fall. if an object falls 16 feet in 1 second, how far will it fall in 3 seconds? how long will it take an object to fall 64 feet? an object will fall in 3 seconds.

Explanation:

Step1: Establish the proportionality equation

Since $s$ is directly proportional to $t^{2}$, we have $s = kt^{2}$. When $s = 16$ and $t = 1$, we substitute into the equation: $16=k\times1^{2}$, so $k = 16$. The equation is $s = 16t^{2}$.

Step2: Find the distance in 3 seconds

Substitute $t = 3$ into $s = 16t^{2}$. Then $s=16\times3^{2}=16\times9 = 144$ feet.

Step3: Find the time when $s = 64$ feet

Substitute $s = 64$ into $s = 16t^{2}$, we get $64=16t^{2}$. Divide both sides by 16: $\frac{64}{16}=t^{2}$, so $t^{2}=4$. Taking the square - root of both sides, $t = 2$ seconds (we consider the positive value of time since time cannot be negative in this context).

Answer:

An object will fall 144 feet in 3 seconds. It will take 2 seconds for an object to fall 64 feet.