Question

1. newton’s second law of motion states that the force applied to an object is the product of its mass and its acceleration, ( f = m cdot a ). write an equation for the object’s acceleration, given the force and the mass.

1. solve for ( w ) in terms of ( a, k, ) and ( t ).

( k = a + 3w - t )

1. a cylinder has a volume of ( 96pi ) ( \text{in}^3 ) and a radius of 4 inches. what is the height of the cylinder?

(hint: the volume of a cylinder is given by ( v = pi r^2 h ).)

Explanation:

Question 1

Step1: Start with the formula \( F = m \cdot a \)

We need to solve for \( a \), so we divide both sides of the equation by \( m \) (assuming \( m
eq 0 \)).

Step2: Divide both sides by \( m \)

\( \frac{F}{m} = \frac{m \cdot a}{m} \)
Simplifying the right - hand side, the \( m \) in the numerator and denominator cancels out, leaving us with \( a=\frac{F}{m} \)

Step1: Start with the equation \( k=a + 3w-t \)

We want to isolate the term with \( w \). First, add \( t \) to both sides of the equation.
\( k + t=a + 3w-t + t \)
Simplifying the right - hand side, the \( -t \) and \( +t \) cancel out, giving \( k + t=a + 3w \)

Step2: Subtract \( a \) from both sides

\( k + t-a=a + 3w-a \)
Simplifying the right - hand side, the \( a \) and \( -a \) cancel out, resulting in \( k + t - a=3w \)

Step3: Divide both sides by 3

\( \frac{k + t - a}{3}=\frac{3w}{3} \)
Simplifying the right - hand side, we get \( w=\frac{k + t - a}{3} \)

Step1: Recall the volume formula for a cylinder \( V=\pi r^{2}h \)

We know that \( V = 96\pi \) and \( r = 4 \). Substitute these values into the formula.
\( 96\pi=\pi\times(4)^{2}\times h \)

Step2: Simplify the right - hand side

First, calculate \( (4)^{2}=16 \), so the equation becomes \( 96\pi=\pi\times16\times h \), or \( 96\pi = 16\pi h \)

Step3: Solve for \( h \)

Divide both sides of the equation by \( 16\pi \) (assuming \( \pi
eq0 \) and \( 16
eq0 \))
\( \frac{96\pi}{16\pi}=\frac{16\pi h}{16\pi} \)
Simplifying the left - hand side, \( \frac{96\pi}{16\pi}=\frac{96}{16}=6 \), and the right - hand side simplifies to \( h \)

Answer:

\( a = \frac{F}{m} \)

Question 2