Parametric
Find dy/dx if
x=t^2+1
y=2t^3-t^2
(6t^2-2t)/(2t)=3t-1
Find the velocity vector is the position vector is
<t^2-1, e^(t^3)>
<2t, 3t^2e^(t^3)>
Two positive integers sum to 100. Find the values of the two integers that will maximize the produce of the square root of the first number times the second.
(sqrt(x))y
x = 33.333
y = 66.667
A cylinder of radius 5 in is being filled with water at a rate of 7in2/min. Find the rate of change of the height of the water.
7/(25pi) = 0.089 (in)/min
Find the tangent line at t=2
x=t^2+t-1
y=t^3-t^2
(y-4)=8/5(x-5)
Find the acceleration vector at t=1 if the position vector is
<t^5-1, 3t^4-2>
<20, 36>
A rectangular plot of land next is 2000ft2 and next to a barn. Find the minimum amount of fencing needed to enclose the plot of land.
126.491 inches of fencing
A radius of a sphere of ice is shrinking at .85 cm/min. Find the rate of change of the volume of the sphere when the radius is 10 cm.
-340pi (cm)^3/min=-1068.141 (cm)^3/min
Find d2y/dx2
x=t^2
y=t^2+6t+5
((2t)2-(2t+6)2)/(2t)^3=-3/(2t^3)
Find the speed of the particle if the vector at t=1 if the position vector is
<t^3+t-1, 4t^2-6t>
sqrt20
A swimmer is in the ocean 6 miles directly from shore. They need to reach their hotel that is 25 miles down the beach parallel to their position. They can swim at 3mph and jog at 5mph. Where should they land on the beach to minimize their time to the hotel?
4.5 miles from their position OR 20.5 miles from the hotel
A cone is point down and has a base radius of 10m and a height of 25m. Sand is draining out of the cone at a rate of 12 m3/min. Find the rate of change of the radius, when the radius of the sand is 5m.
0.0611 m/min OR
-0.192/pi
Find the points of the horizatonal tangents if t is all real numbers
x=t^2-t+1
y=t^3-3t
t = -1 (3, 2)
t = 1 (1, -2)
Determine if following particle is moving right or left at t=2 and justify your answer if the position vector is
<t^2-6t, t^3-7t>
left since dx/dt is -2
An open topped box has a length that is 4 times the width. If the box holds 400in3, what are the dimensions of the box to minimum amount of material needed to construct the box?
5in by 20in by 4 in
-7.647 mph
Find the points of the vertical tangents if
0<=t<2pi
x=3+2cost
y=-1+4sint
t = 0 (5, -1)
t = pi (1, -1)
Given the following position vector, decide if the particle is ever at rest and explain your answer.
<1/3t^3-4t+6, 1/3t^3-4t^2+12t-8>
Yes since both dx/dt and dy/dt =0 when t=2
A cylinder is inscribed in a sphere that has a radius of 8 cm. Find the maximum volume of the cylinder.
394.138pi cm^3= 1238.220cm^3
A sphere of ice is melting at into a cylinder below that has a radius of 9 inches. The radius of the sphere is shrinking at a rate of 0.54 in/sec. Find the rate of change of the height of water in the cylinder when the radius of the sphere is 12in.
3.84 in/sec