ATTRACTION VERSUS REPULSION Consider the motion of a particle in one
dimension (x-axis) under the simultaneous influence of two forces. One force repels the
particle from the location x 1, while the second one attracts the particle towards the
location x = -1. In both cases, the force is inversely proportional to the distance from x = 1 and x = -1 respectively, but the proportionality constants may be different. Describe the trajectory of the particle over time, for different values of the two proportionality
constants. Does the motion of the particle change qualitatively as you vary the proportionality constants?

• Define variables that capture the important components of the situation.
• Set up equations to describe the desired relationship between the variables you have defined.
• Describe how to solve the equations you set up.
• Interpret the solutions you obtained in the context of the original situation.

it is true

step-by-step explanation:

all the combinations add up i used   calculator

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.

hard

step-by-step explanation: