Particle Motion
The velocity of a particle and the position of a particle have this relationship.
What is velocity is the derivative of position ?
This is the equation for Newton's Method.
What is Xn+1 = Xn - [ ( f(Xn) ) / ( f'(Xn) ) ] ?
Jerry wants to construct a cylindrical can with a bottom but no top to hold his revolutionary new sports drink, Jerry Juice. The can will have a volume of 355 mL (1 mL = 1 cm3). These can dimensions will minimize the amount of material needed to construct the can.
r = (710 / 2PI)1/3 = 4.8346
h = 355 / ( PI * ( 710 / (2PI) )2/3 ) = 4.8346
This is dY/dX of Y2 - XY = 3 .
What is dY/dX = (3 + Y) / (2Y - X) ?
This is the function a2 + b2 = c2 derived with respect to time.
a(da/dt) + b(db/dt) = c(dc/dt)
This is the equation for the velocity of a particle whose position is given by the equation s(t) = 32t2 - 16t + 32.
What is v(t) = 64t - 16 ?
This is X2 given from using Newton’s Method for f(x)=x3−7x2+8x−3 if X0=5
What is X2 = 5.71875 ?
These are two positive numbers, x and y, that make x+2y=50 and (x+1)(y+2) a maximum.
x = 53/2
y = 47/4
This is dY/dX of X = Y3 + 37Y .
What is dY/dX = 1 / (3Y2 + 37) ?
A Jerrybird is flying 6 myriameters above the ground in Giant Land. The Laboratory of Giant Bird Research determines that the distance between the station and the Jerrybird is 10 myriameters and that this is changing at a rate of -400 myriameters per hour. What is the rate of change of the distance between the station and the point directly below the Jerrybird?
What is 500 miles per hour ?
This is the equation for the acceleration of a particle whose position is given by the equation s(t) = 32t2 - 16t + 32.
What is a(t) = 64 ?
This is an approximation of (5)1/2 using Newton's Method and twoiterations starting with X0 = 2.
What is X2 = 2.23611 ?
Assuming Jerry has 3000 feet of fencing to surround his water buffaloes, this is the maximum area that Jerry can encompass with his fencing.
What is 562500 square feet?
(Length = 750 and Width = 750)
This is dY/dX of the function X2 + Y3 = 4.
What is dY/dX = (−2X) / ( 3Y2 )?
This is the rate of change of the radius and the rate of change of the surface area of a balloon that is being pumped with 4.5 cubic feet of air per minute when the radius is 2 feet.
What is
dr/dt = 4.5 / ( 16 * PI ) feet per minute
dSa/dt = 4.5/PI feet per minute
?
This is the acceleration at time t = 7 of a particle whose position is given in meters by the equation s(t) = 32t2 - 16t + 32.
What is a(t) = 64 (meters / second2) ?
This is the approximation of X3 of X7 + 4 = 0 using Newton's Method starting with X1 = −1.
What is -16.9028 ?
Jerry wants to build a box that has base length 6 times the base width and that will enclose 20 cubic inches. The cost of the material of the sides is $3 per square inch and the cost of the top and bottom is $15 per square inch. These box dimensions will minimize the cost.
w = (7/18)^(1/3) = 0.7299
l = 6w = 4.3794
h = 10 / ( 3(0.7299)^2 ) = 6.2568
This is the equation of the line tangent to X4 + Y2 =3 at the point (1,−√2).
What is Y = (2)1/2 * (x−2) ?
This is the instantaneous rate of change of the height of a cone whose volume is increasing at a rate of 2 meters3 / second and which has a radius of 3 meters when the height is 2 meters.
What is dh/dt = 1 / (18*PI) (meters / second) ?
This is the velocity at time t = 4 of a particle whose position is given in meters by the equation s(t) = 16t4 - 13t3 - 12.
What is v(t) = 3460 (meters / second) ?
This is the approximation of 1001/100 using Newton’s Method starting with X1 = 1 and going to X4.
X4 = 1.950399
Jerry has 45 square meters of chocolate bar with which to build a chocolate box with a square base and no top. These are the dimensions of the box that will maximize the enclosed volume.
What are
l = w = √(15) = 3.8730
h = 45−154√(15) = 1.9365
?
This is the equation of the line tangent to
Y2e2X =3Y + X2 at (0,3).
What is Y = −6X + 3 ?
The Jerrycat is a breed of giant cat in the mythical land of giants. One Jerrycat that is 6 meters tall is standing X meters away from a light post. The post is 15 meters tall. As the Jerrycat moves away from the light post at a rate of 3 meters per second, this is the rate of change of the length of that Jerrycat's shadow.
What is 2 meters per second ?