Question

write a formula for a linear function that models the situation. state what the input variable represents and the domain of the function. a stone is dropped from a water tower and its velocity increases at a rate of 32 feet per second. the stone hits the ground with a velocity of 256 feet per second. write the formula for v(t) v(t) = 32t what does the input variable represent? velocity distance time acceleration state the domain of the function a d = {t | ≤ t ≤ } b. d is all real numbers

Explanation:

Part 1: Formula for \( v(t) \)

Step 1: Identify the type of motion

The stone is in free - fall, and its velocity increases at a constant rate (acceleration due to gravity, here \( 32\) ft/s²). The velocity of an object in free - fall starting from rest (\(v_0 = 0\)) is given by the formula \(v(t)=v_0 + at\), where \(v_0\) is the initial velocity and \(a\) is the acceleration. Since the stone is dropped, \(v_0 = 0\) and \(a = 32\) ft/s².

Step 2: Derive the formula

Substituting \(v_0=0\) and \(a = 32\) into the formula \(v(t)=v_0+at\), we get \(v(t)=0 + 32t=32t\).

In the function \(v(t) = 32t\), the input variable is \(t\). In the context of a stone being dropped and its velocity changing over time, \(t\) represents the time (in seconds) that has elapsed since the stone was dropped.

Step 1: Find the time when the stone hits the ground

We know that the final velocity \(v = 256\) ft/s. Using the formula \(v(t)=32t\), we set \(v(t)=256\) and solve for \(t\).
\(32t=256\)

Step 2: Solve for \(t\)

Divide both sides of the equation \(32t = 256\) by \(32\): \(t=\frac{256}{32}=8\) seconds.
Since time starts at \(t = 0\) (when the stone is dropped) and goes up to \(t = 8\) (when the stone hits the ground), the domain of the function \(v(t)\) is the set of all real numbers \(t\) such that \(0\leq t\leq8\).

Answer:

\(v(t)=32t\)

Part 2: What the input variable represents